SOLUTIONS OF S I M U L T A N E O U S O R D I N A R Y
SINGULAR
DIFFERENTIAL
EQUATIONS.
B Y C. N. SRINIVASIENGAR, D.Sc. Central College, Bangalore. R e c e i v e d D e c e m b e r 3, 1934. ( C o m m u n i c a t e d b y P r o f . ]3. V e n k a t e s a c h a r , .~t.A., F.tnst.P.)
[w 1].
Introduction.
IN considering a system of ordinary simultaneous differential equations, it is sufficient to regard the equations as involving differential coefficients of the first order only; for, equations involving differential coefficients of higher orders can be easily reduced to this canonical form by introducing more variables. We shall mostly confine ourselves to equations involving three variables only, so that the equations m a y be taken in the form r 1,2) . . . . . . (1) where y'
dy
z' --
dz
That such a pair of equations can possess singular integrals* is well known, and when r and r are taken in certain particular forms, methods of obtaining the singular solutions are known. A list of chief references is given at the end. The subject will be studied in this paper by adopting a different method of procedure, which enables the singular solutions to be determined even when r and r involve elementary transcendental or irrational expressions. [w167 2-4]. General Solutions of the Equations (1). w2. Let the equations (1) be solved for y' and z', giving continuous functions of x, y and z possessing continuous first partial derivatives. [A] * T h e defiuit.ion of Singular Solutions a d o p t e d is t h a t t h e y are s o l u t i o n s n o t included in the general solutions. T h e l a t t e r should, h o w e v e r , be c a r e f u l l y defined b y the un~,lueness p r o p e r t y in t h e f u n d a m e n t a l e x i s t e n c e t h e o r e m s of C a u c h y . V~de m y p a p e r in the Journal of the Indian Mathemafical Society ( J u b i l e e Volume, Vol. 20), also in t h e Half-Yearly Journal of the Mysore University, Vol. V. ~f T h e only a t t e m p t in this d i r e c t i o n so far is b y M. J . M. ZIill who in a p a p e r {Ref. 1) o n partial differential e q u a t i o n s has discussed some e x a m p l e s of t h e t y p e dx = d_z _-- --.dr His m e t h o d s are, h o w e v e r , v e r y t e n t a t i v e . I h a v e f o u n d , h o w e v e r , his P Q R e x a m p l e s v e r y i n s t r u c t i v e a n d useful in t h e p r e s e n t m e t h o d of p r o c e d u r e , a n d s e v e r a l of his e x a m p l e s h a v e b e e n used in this p a p e r .
668
S i n g u l a r Solutions o / D i f f e r e n t i a l .Equations
669
Let
y ' = ~1 (x,y, z)
. . . . . . . .
z' = ~b2 (x, y, z)
.
Hence, y~ =
~~ i
y, ~~v r
+
+
.
.
r ~1
. .
. .
. .
. .
.
(2)
.
(3)
.
(~)
I,et z be eliminated between (2) and (4), so as to result in a differential equation of the second order in x and y, in which the coefficients are all continuous functions. For this, (2) must be solved for z, and the value substituted in (4). [B] :Let the resulting equation be written f (y", y', y, x) = 0 . . . . . . . . The equation (5) possesses two independent first integrals of the type F~ (x,y,y',c~) ---- O, ( r = 1,2) . . . . . . Substituting y ' = r the general solutions~, of (1) are obtained 9
F~ (x,y, r
c~) = o,
(r = 1, 2)
(5) (6)
. . . . . .
(7)
. . . . . .
(8)
Finally, (7) m a y b e p u t into the form u (z, y, z) = c l ;
v (x, y, z) = c~
w 3. [The above general method of solution has one important d e f e c t ~ a defect which cannot be removed. We are led to believe t h a t the general solutions of any given equations (1) can be completely represented in t h e form u = cl ; v = c2, or in the rationalised form F , (x, y, z, c,) = 0, (r = 1, 2), each of the F's involving only one constant. This is not necessarily the fact. Consider for example the generalised Clairaut-equations r ( y - x y ' , z - r e ' , y', z') = 0, (r = 1, 2). Differentiating and solving, we get y" = 0 = z", showing the integral curves (other t h a n singular solutions) must necessarily be straight lines. B u t t h e general method of w 2 will not give straight lines only. (5) should necessarily reduce t o y " = 0, whose first integrals a r e y ' = a, and y - - x y ' = b. Consider a concrete example : y =
x y ' + y , 2 + z' ; z =
xz' + y ' z ' .
Eliminating a and b from the known general solutions y = a x + a 2 + b ; z = bx+ab, we get a (x + a) 2 - y
(x + a) + z =
0
.
.
.
.
.
.
(9)
bS-- b2y - bzx + z~ = 0 . . . . . . (10) The function ~bl (x, y, z) of (2) is evidently the same as that obtained by solving (9) for a. Hence the equation u = ci in (8) gives (9) on rationalisation. Similarly v = cz leads to an equation of degree higher than t h e first.
$ Forsyth~ T~'eati~e on D~)~eren~ial Efluat'~ons ~ w
670
C. N. S r i n i v a s i e n g a r
The general solutions (8) will thus represent curves of a degree higher t h a n the first. They must necessarily be all degenerate, breaking up into straight lines which represent solutions of the given differential equations, and other curves which are not solutions.* Assuming that there exist for the equations (l) general solutions which are completely representable in the form F~. (x, y, z, a, b) = O, r = 1, 2,~ the solutions might only be partially re#resented when F1 = 0 and F~ = 0 are transformed into some other form. The explanation for the appearance of the extraneous curves seems to be as follows" In geometry, the curve given by the equations $~ (x, y, z, a, b) = 0, r = ] , 2 is known to form only a part of the curve common to the surfaces obtained by rationalising the equations %hi = a and $2 = b (continuing the notation of w 2). Similarly we must regard the equations (2) and (3) as representing something more t h a n the original equations (1). If, however, we start with the equations (2) and (3), (8) will completely represent their general solutions. Thus, when we deal with equations of t h e dx dy dz type -~ = ~ = . ~ , the general solutions are completely representable in t h e foITn
~
-~
C1 ;
"/) =
C2 .
We shall hereafter be considering the general solutions of (l) in t h e form u = c l ; v=c2 with the understanding that this includes the general solutions though not of necessity completely equivalent to them.3 w4. The equations (1) can have only one set of independent integrals. If u = c l ; v=c2 be such a set, the general solutions may also be supposed to be included in general in 81 (u, v) = c1 ; 82 (u, v) = c9, where 9i and 82 m a y be any arbitrary functions, subject to suitable limitations. These facts are easily proved, and are written down here only for t h e sake of reference. [w167 5-9~. Equations leading to Singular Solutions. w5. Our object is to detemfine a pair of equations satisfying the given differential equations, which are not included in the general solutions (8) for any values of cI and c2 whatever. At least one of the two equations to be determined must be functionally different from u and v. Only one of t h e equations may be different or both, and accordingly two classes of singular solutions might exist. * A. ~'Iayer (Ref. 4 ) e x p r e s s e s ~he solutions in t h e f o r m Y=q~l(~, a, b}; z---a, b)--a f o r m possessing similar l i m i t a t i o n s . Also Ref. (7), p. 32~. t I t is however possible t h a t solutions m a y exist o n l y i n t h e f o r m x = 91 (t, a, b) ; y 92 ( t , a , b ) ; z = S s ( t , a , 5), t being ~ p a r a m e t e r .
r
Singular Solutions o/ Differential ~'quations
671
Assuming the existence of such sing,,dar solutions, it follows that their existence must be accounted /or by the f~ct that the general theory ofw can break down somewhere. Now this may happen in either of the two places m a r k e d in w as [A] and [B]. I n order t h a t the theory m a y break down in the place marked [A], i.e., in order t h a t it m a y not be possible to obtain from (1), unique solutions y ' = r ix, y, z), z ' = r ix, y, z) in the form of continuous functions with continuous first partial derivatives, it is necessary t h a t either
~ (r 6_~) _ 0, . . . . (11) NY', z') "" or ~1 or r or a n y of their first partial derivatives w i t h respect to x, y or z = infinite (or indeterminate)* . . . . . . (12) Conversely, if a function wl ix, y, z) ----0 is constructed b y means of values of x, y and z satisfying (11) or (12), wl = 0 when t a k e n along w i t h a n o t h e r suitable equation, may represent a solution of the given differential equatious and we m i g h t expect these solutions to be singular. I n order t h a t the general theory m a y b r e a k down~ in t h e place m a r k e d [B] in w2, i.e., in order t h a t the result of eliminating z between (2) and (4) m i g h t involve coefficients or functions which can admit of discontinuous derivatives for finite values of the variables, either it must n o t be possible to solve (2) for z in the form of a unique continuous f u n c t i o n of x, y, y ' w i t h continuous derivatives, or otherwise t h e first derivatives of r m a y not be continuous for all finite values of x, y, z. We can write as necessary conditions, either 5 r
0,
.
.
.
.
.
.
(13)
or r or r or any of their first partial derivatives = infinite (or indeterminate) . . . . . . . . . . (14) The condition (14) however resolves itself into the conditions (11) and (12), so t h a t (13) is the only new condition t h a t presents itself. We shall 9. ~r however prove in w 9, that the e q u a u o n - ~ = 0, call it we ix, y, z) = 0, does not give rise to any true singular solutions, unless wo. is included in Wl. * This follows from the theory of implicit functions. V~deGoursat-Hedrick, Math. Analysis, Vol. I, Chapter lI. The remarks of Hedrick in the footnotes deserve attention. ~f The break-down of the theory means that the conditions for Cauchy's exisbenee theorem are not satisfied for (5). The existence of two and only two independenb first integrals of this equation is an immediate corollary of the uniqueness of the general primitive as giveu by Cauchy's theorem. The possibility of existence of any " singular first integrals " depends upon the break-down of Cauehy's theorem as regards uniqueness.
672
C. N. Srinivasiengar
w 6. L e t w3 (x, y, z) = 0 d e n o t e a n y o n e of t h e f u n c t i o n s i n v i r t u e of which u, or v, or a first p a r t i a l d e r i v a t i v e of u or v b e c o m e s infinite. w 7.
F r o m t h e g e n e r a l s o l u t i o n s u = Cl, v -----cs, w e obtain
y'=~b~(x,y,z)
=~(z,x).
~ y , z)
z' = r
= ~(u'v)'~(~'~)
(x, y) " b (y, z)
T h e f u n c t i o n wl m a y t h e r e f o r e arise o u t of a n y of t h e following causes : (i) a first d e r i v a t i v e of r or r m a y b e c o m e infinite w h e n a first d e r i v a t i v e of u or v b e c o m e s infinite, i.e., wl = 0 m a y b e i n c l u d e d in ws = 0 ;
(ii) a first d e r i v a t i v e of r or ~b2 m a y b e c o m e infinite w h e n a s e c o n d d e r i v a t i v e of ~ or v b e c o m e s infinite ; (iii) @l or r or a first d e r i v a t i v e of e i t h e r m a y b e c o m e infinite i n v i r t u e of ~ (u, v) = O, i.e., wl m a y b e a f a c t o r of ~z -(u, (y, z) -(y, - - ~v).'
(iv) ~b1-1 or ~b2-1 or a d e r i v a t i v e of e i t h e r m a y b e c o m e infinite i n _ _ _ v i r t u e of b (u, v)
(z, $}
0 or
~ (u, v)
~ (x, y)
-- 0.
W e can h o w e v e r p r o v e t h a t unless (i) be t h e case, wl = 0 will n o t c o n t r i b u t e t o w a r d s a n y s i n g u l a r solutions. W e shall first consider a l e m m a . w :Let f (x, y, z) = 0; g ( x , y , z ) = 0 d e n o t e g i v e n solutions of t h e differential e q u a t i o n s w h o s e g e n e r a l s o l u t i o n s are u = cl ; v = c2. I t is a s s u m e d t h a t f and g are a n a l y t i c for all finite v a l u e s of t h e variables, a n d m a y be m a d e to r e p r e s e n t p r o p e r surfaces. I t is also a s s u m e d t h a t u a n d v are finite w h e n f = 0 and g = 0. I f e i t h e r of t h e m b e c o m e s infinite w h e n f = 0 a n d g = 0, it m u s t be r e p l a c e d by its reciprocal. T h e n we h a v e : T h e necessary a n d sufficient c o n d i t i o n s t h a t f = 0; g = 0 s h o u l d b e i n c l u d e d in t h e general s o l u t i o n s are t h a t t h e limits of t h e e x p r e s s i o n s (u, f, g) ~ (v, f, g) s h o u l d be both e q u a l to zero. Cx, y, z) ' ~ (x, y, z) A proof is easily s u p p l i e d b y following t h e m e t h o d a d o p t e d for t h e case of t w o variables. [Proc. L.M.S., Vol. 17 (Series 2), pp. 159-161.] W e s h a l l express this result in a m o r e c o n v e n i e n t form. L e t ~ (x, y, z) -- ~ x
~yMl-t-
~ z N 1 = A,
a n d b (v, J, g)
~g
bg
~g
so t h a t A a n d B t e n d t o zero as f--> 0, g--> 0. H e n c e 5g ~g (I,s~fl--LiMs) + (L21~I - LIi~72) = AL2 - BL~ -> 0,
673
S i n g u l a r Solutions o f Differential Equations i.e., e x p a n d i n g ,
II 5g
W e g e t t w o similar equations b y e l i m i n a t i n g ~-~ or
bg
bz'
Hence,
f a n d g b e i n g i n d e p e n d e n t , we m u s t h a v e f--> O, L t g -+ 0 b (u, v,/3 _ O. (x, y, z) S i m i l a r l y f -+ O, L t g -> 0 b (u, v, g) = O. The s a m e m e t h o d gives t h e f o r m e r "
b ix, y , z)
pair of e q u a t i o n s ,
b (u, 1, g)
5 (v, 1, g)
viz., b (x, y, z) -> O, b (x, y, z) -> O, from t h e latter.
THEOnF,~ 1. The necessary and sufficient condition that f = O, g ---- 0 should be singular solutions is that the limiting value of at least one of the b (u, v, f) ~ (u, v, g) expressi~ b (x,y, z)' ~ (x, y, z) should be distinct from zero, when f---> O, g---> O. If o n l y one of t h e s e limits b e different from zero, t h e solutions are c a l l e d singular solutions of the first order. Singular solutions of t h e second order also m a y exist, for which, however, t h e limits of all t h e four determ i n a n t s (u x v r fz), (ux v r g~), (ux f r gz), (vx f j, g..) should be d i s t i n c t f r o m zero. T h e conditions that the solutions f = O, g = 0 might be particular are
t h a t L t * b ( u , v , f ) _ 0 and I,t b ( u , ~ , g ) _ (x, y, z) ~ (~, y, z) v i r t u e of t h e e q u a t i o n s f = O, g = O.
0 either identically,
or in
Illustrations :
]4XA-~PLE I. To t e s t t h e n a t u r e differential equations
~b
of t h e solutions z - - x - - y = O; z - 2 y = b
dy
l + ( z _ m _ y ) 8 9 = ~- =
of t h e
dz
~.
Here, u = y + 2 (z--x--y) 89," v = z - 2y. H e n c e , t a k i n g f = z--x,--y, Lt b (u, v, .f) b (y, z--2y, z--x--y) f --> O, g --> 0 ~ (x, y, z) = ~ (x, y, z) =/= O. T h e solutions are t h e r e f o r e singular solutions of t h e first order. * The possibility of some of the first derivatives of u or v becoming infinite in virtue of f = O, g = 0 is allowed. Compare Mysore Un~l:r Journal, Vol. u
C. N. Srinivasiengar
674
EXASMPLE 9.. To t e s t t h e n a t u r e of t h e solutions z = x + y ; x -~ -- y2 = c o n s t a n t , of the equations
av x (z-~-y) 89 - y
y (z-x--y) 89 - x dz X
2(z-x--y) 89 + (x+y) ( z - x - - y ' - - l ) H e r e , u - x + y (z--x--y) 89 v =- y + x (z--x--y) 89 W r i t e f ~- z - x - y ;
g - x2--y~
Lt ~ (u, v, .f) _ 1 9 Lt ~ (u, v, g) 0. T h e n , ] _ > 0, g ---> 0 5 (x, y, z) ' f--> 0, g --> 0 ~ (x, y, z} = T h e solutions a r e t h u s s i n g u l a r a n d of t h e first o r d e r .
]~XAMPLE 3. To t e s t t h e n a t u r e of t h e s o l u t i o n s f the equations
dx dy y2 (z--x--y) 89 -- y (y--2x) = y~ Here, u - x + y
z - - x - - y = 0 ; g = x - - c = 0 of
dz (x--y) ( z - x - y ) +y 2 ( z - - x - y ) 89
(z--x--y) 89 v =_ x 2 + y ~ ( z - - x - - y ) 89.
Lt ~ (u, v, f) f---> 0, g --> 0 ~ (x, y, z)
(x, x'-, z - x - y ) (x, y, z)
Lt ~ (u, v, g) = ~ (x, x ~, x - c ) J - + O, g -~ 0 ~ (x, y, z) 5 (x, y, z)
=
_ O. O.
T h e g i v e n s o l u t i o n s s h o u l d t h e r e f o r e b e particular. T h i s is e a s i l y verified. T h e g e n e r a l i n t e g r a l s are also g i v e n b y v - - u 2 = c o n s t a n t ; u = constant. (w 4). B u t v - - u 2 c o n t a i n s ( z - - x - y ) 89as a factor. T h e s o l u t i o n s f---- 0, g ----- 0 are t h u s included i n t h e g e n e r a l i n t e g r a l s t a k e n in t h e f o r m v - - u 2 ----- c o n s t a n t , u = c o n s t a n t . w 9. I t follows i m m e d i a t e l y f r o m T h e o r e m 1 t h a t w 1 = 0 will n o t c o n t r i b u t e t o a n y s i n g u l a r s o l u t i o n unless it is i n c l u d e d in wa ---- 0. F o r , t h e e q u a t i o n s (2) and (3) are e q u i v a l e n t t o ~u ~-~ dx + ~~u d y + ~~u- d z = O ;
~~v- x d X +~v ~ d y + ~byd z = O .
I f wl = 0 c o n t r i b u t e s t o w a r d s a s i n g u l a r s o l u t i o n ,
~wl 5x dx + ~wl -@ dy + bwl -~z dz = 0 m u s t b e satisfied b y t h e v a l u e s of y ' , z' o b t a i n e d f r o m t h e above.
T h e J a c o b i a n ~ (u, v, w) c a n n o t a t t a i n a n o n (s, y, z) zero v a l u e , m | e s s one or m o r e of t h e d e r i v a t i v e s of u or v b e c o m e s i n f i n i t e in v i r t u e of wl = 0.
S i n g u l a r Solutions o[ Difl'erenliM Equations
675
A similar argument holds to show that if w~ = 0 leads to a singular solution, we is included in we, i.e., wo. = 0 makes one of the first derivatives of u or v infinite, and from the values of ~bl and ~.2 in terms of u and v we can conclude t h a t we is included in wl as well. W e conclude t h a t in order to obtain the singular solutions from the differential equations, it is enough if we consider the function defined by (11) or (12). I n order to obtain them from the general integrals taken in the form u = cl, v = co., we must consider the function w:~. The equation wl = 0, or w3 = 0, as the case m a y be, must be associated with a suitable other equation whose formation will be now studied.
[w167 10-11].
Determination of the equation to be associated along with w 1 = 0 or w3=O. w For a particular method of formation of Wl = 0, the question will be found discussed in Forsyth. (Ref. 3, pp. 150-133; also Ref. S.) The result will be quoted here, with the symbols altered: "' If values of y ' and z' satisfy the equations (i) 61 (x,.y, z,y', z') = 0 (ii) ~2 (x, y, z,y', z') = 0 (iii)
0 =
(iv)
0 = b (61, $~) + y, ~ (r r (x, z') ~ (y, z')
(v)
~ (r
0= j =
r
y')
(r
+ y ' 8 (r
r
(v, v')
+ z' ~ (r
r
Cz, v')
+ z' ~ (6~' r ~ ~z, ~')
,
then the equations $1 = 0, ~.2 = 0 possess a singular, solution involving an a r b i t r a r y constant ; this singular solution is constituted by the combination of the general integral of I-I (z,y, y ' ) = 0 which is the eliminant in z and z' between ~1 = O, r = O, J = 0 with wj (x, y, z ) = 0 which is the eliminant in y' and z' of the same three equations. The theorem m a y be divided into two parts ; the second part given by t h e italics denotes the method of obtaining the singular solutions (supposed to exist) from (i), (ii) and (v); the first part gives (iii) and (iv) as the conditions of sufficiency in order t h a t the equations (i), (ii) and (v) might give a set of solutions (singular or otherwise). The first p a r t is the analogue of t h e familiar theorem. " If a singular solution of the equation r y, p) ~- 0 exists, it must satisfy the equations $ = 0 ; ~-r -- 0"
+ p
= 0 simul-
taneously." I t is now known that this theorem is not necessarily true if $ is considered as involving irrational or transcendental expressions. The
676
C. N. S r l n i v a s i e n g a r
theorem should be recast as follows : " If the singular solution satisfies
--=-0 and ~p -- O, it must also satisfy 54~-~H-P ~y = 0. "
Similar remarks
hold in the present connection. w II. In the second part of the theorem, the equation to be taken along with wl = 0 so as to constitute the singular solutions is obtained b y eliminating z and z' between ~i = 0, ~2 = 0 and ] = 0, or what is the same thing, between ~, = 0, r = 0 and w = 0. This latter form of expression is more advantageous, since this does not take cognisance of the particular method of formation of the equation w = 0 ; in other words, w m a y be u'1 or w~. (We have of course seen t h a t if w is wl, it is also ws, and vice versa, if w = 0 could lead to singular solutions.)
The following is a n o t h e r w a y of obtaining the second equation, and is of considerable significance, t h o u g h it is theoretically equivalent to t h e above mode of getting H (x, y, y') = 0. F r o m t h e equations (1), we get in general y ' = r (x,y, z) ; z' = #.~ ( x , y , z). These l a t t e r equations go together when we are considering a n y solutions which are particular cases of t h e general integrals of the given equations. L e t us now consider one of these equations along with w = 0, where w means either wl or w3. Thus,
w = O; y" = #~ (x, y , z )
. . . . . .
(15)
or w = O ; z ' = r . . . . . . (16) (15) or (16) may be considered as a new pair of simultaneous equations. If any solution of (15)also satisfies (16)or vice versa, it satisfies (2) a n d (3) as well. Since singular solutions of (2) and (3) can be generated only t h r o u g h the functions wl or wa, it follows t h a t such singular solutions are all given by (15) or (16). I t is also evident t h a t singular solutions of (2) and (3) that are given by (15) are equivalent to those given by. (16), for a n y such solution should satisfy b o t h (15) a n d (16). Let (15) give on elimination of z and solution for This process is equivalent to t h a t of eliminating z between w = 0. I n other words, the equation y ' ---- A(x, y) H ( x , y , y ' ) = 0, except t h a t w is t a k e n in its more general let (16) give z' = / ~ (z, x).
y', y ' = A (x, y). $l = 0, ~2 = 0, is e q u i v a l e n t to form. S i m i l a r l y
The general integral of y ' = A (x,y), or z' = / ~ (z, x) when t a k e n along with w = 0 may constitute a set of singular solutions of the given e q u a t i o n s . The singular solution* of y ' = A, or z' = /~, if a n y exist, when c o m b i n e d * The singular solution of y' = ~, if any exist, is ~ven by -bx- -- infinite, or ~ = infinite. Vide my paper in Tohok.~Mathematical Journal, Vol. 30.
677
S i , zffular Solutions o f Di/~erentiat Equations
w i t h w = 0 also may c o n s t i t u t e solutions of (1). T h e l a t t e r solutions, if n o t i n c l u d e d in t h e general solutions or in t h e singular solutions of t h e first order, will be singular solutions of the second order. [w 72].
Simplification in the case where the two systems of surfaces u = a and v = b do not possess a common envelope.
w 12. W h e n t h e general integrals u = a, v = b are known, we shall a r r i v e a t a rule w h i c h enables us to write d o w n t h e solutions of (15) or (16) s t r a i g h t - a w a y in m o s t cases. I n m o s t cases, t h e e q u a t i o n wa = 0 m a k e s t h e first p a r t i a l d e r i v a t i v e s of only one of t h e f u n c t i o n s u or v infinite. L e t us s u p p o s e t h a t w3 --- 0 m a k e s one or m o r e of t h e first d e r i v a t i v e s of u infinite. Also, u itself can be t a k e n as not b e i n g infinite w h e n w3 = 0, for otherwise, u m a y be replaced b y u -1.. L e t D ( x , y , z ) = c o n s t a n t b e t h e equatioxl wtlich w h e n t a k e n along w i t h ws = 0, gives singular solut i o n s of the first order. F r o m ~ 8, it follows t h a t w3--->0 Lt ~~ (u, wa) =/= 0, b u t (x, v, y,z) (u, v, ~ ) -- 0, so t h a t ~ is a f u n c t i o n o f u a n d v , say~ =O(u,v). Now, (x, y, z) w3 ----- 0 e n v e l o p e s t h e s y s t e m of surfaces u = c o n s t a n t , and unless Q is i n d e p e n d e n t of u, also t h e s y s t e m D (u, v) = constant.~ If therefore ~ is n o t i n d e p e n d e n t of u, t h e curves w3 = 0 ; ~ = c o n s t a n t w~ll be t h e c h a r a c t e r istic c u r v e s on t h e envelope of 52 ---- c o n s t a n t . W e shall now prove t h a t s u c h c u r v e s c a n n o t r e p r e s e n t singular solutions at all for t h e g i v e n differential equations.
To begin with,
let f ( x , y, z, a) = 0 be a s y s t e m of surfaces w h e r e f i n v o l v e s n o i r r a t i o n a l or t r a n s c e n d e n t a l expressions, so t h a t t h e e q u a t i o n of t h e e n v e l o p e E is o b t a i n e d in t h e usual way, by t h e e q u a t i o n s f = 0,
baf - - 0. I t m u s t be n o t e d t h a t the d~rection cosines of the tangent at any point on a characteristic curve cannot be obtained directly from the equations
if, E) ~(f,E)
~(J,E)
f = 0, F, = 0 ; for t h e expressions ~ (y, z~ ' b (z, x) ' ~ (x, Y) are all e q u a l t o zero.
T h e d i r e c t i o n cosines will h a v e to b e o b t a i n e d from t h e e q u a t i o n s
f = O, ~~-~-f -- O, i.e., t h e y are p r o p o r t i o n a l to ~ ~ f '(y,g-a z) , etc., etc. N o w considering t h e f o r m u -- a, and its e n v e l o p e E = ws = 0, t h e same thing happens.
The expressions ~~ ((u, y,zw ) 3) ' ~~ ((u, z , x w~) ) '
b~ ((u, x , y w~) ) are
~[ For proof, see " E n v e l o p e s of S y s t e m s of SurEaees ", ToholcuMath. Journal, Vol. 39.
678
C. N. S r i n i v a s i e n g a r
either all equal to zero, or some of t h e m are i n d e t e r m i n a t e . To obtain t h e directions of the t a n g e n t at a p o i n t on the characteristic, the equation u = a must first of all be reduced or simplified b y the help of w3 = 0. T h e expression u (which is itself finite when w~ = 0) involves a fractional power of w3, or involves logarithmic or other t r a n s c e n d e n t a l functions of w3. L e t u ' = a be the equation o b t a i n e d b y simplifying u = a++, with the help of w3 = 0. (For instance, in E x a m p l e 2, w3 =-- z - - x - - y ; u' - x.) Then u' = a, w3 = 0 are the equations of the characteristic curve. If these equations should satisfy ~ (~'' y,u, ~)v) = 0. But u' is a f u n c t i o n - the differential equations, b-(x, of u and w3, so t h a t it follows t h a t w3 m u s t be a function of u and v. w3 = 0 cannot therefore give rise to singular solutions of the first order. In o t h e r words, the characteristics u' = a, w3 = 0 can at the most give a set of solutions included in the general solutions. (See E x a m p l e 11 for an illustration of this extreme case.) I n no case can t h e y give singular solutions. We conclude t h a t 52 is i n d e p e n d e n t of u. T h e singular solutions g i v e n 3y w~ = 0 ; .y' = r (x, y , z) are therefore equivalent to w3 = 0 ; v (x, y, z) = b. (We assume t h a t w3 = 0 is n o t included in v = b, for any value of b.) A purely geometrical m e t h o d of arriving at this result is considered in w
We have assumed t h a t w3 = 0 makes the first derivatives of u o n l y , infinite. If w3 = 0 is also the envelope of the system v = b, the above a r g u ments do not hold. In such cases, sometimes it m a y be possible to o b t a i n a function $ (u, v), none of whose first p a r t i a l derivatives becomes infinite when wz = 0. ( V i d e E x a m p l e 9.) The singular solutions are t h e n g i v e n by w3 = 0; $ ( u , v ) = b. If, however, it is n o t possible to find such a function, the actual solution of t h e differential equations y ' = $l (x, y , z) ; ~3 = 0 cannot be avoided. [w 731. R e m a r k s . w 13. The methods of derivation of the singular solutions of the first order have been studied. So far, we h a v e been dealing only with t h e necessary conditions for singular solutions. A n y singular solutions t h a t m a y exist will be derivable b y the foregoing methods, b u t there is no g u a r a n t e e t h a t the equations derived from these methods do always actually c o n s t i t u t e solutions., whether singular or particular. For t h e special case wherein t h e equation wl = 0 is formed by tile elimination of y ' and z' between $1 --- 0, $9 = 0, J = 0, the t h i r d and f o u r t h equations in the theorem quoted in w 10 give the conditions of sufficiency in order t h a t t h e equations obtained b y I t is i m m a t e r i a l w h e t h e r w e u s e ~ = a, or ~
(~,v) = constant.
Singular Sohtlions of Differential Equations
679
the process described in the theorem might constitute solutions (singular or particular). In other cases it will be perhaps simplest to make a direct verification of the fact whether the differential equations are satisfied. The t h e o r e m of w 8 m a y often be useful in deciding whether the solutions o b t a i n e d are singular or are included in the general integrals.
[w167 14-15 I.
On Singular Solutions of the Second Order.
w 1-1:. We shall now examine the implication of the existence of singular solutions of the second order. Consider one of the fmmtions which goes by the name w3. We shall assume t h a t for this w3, the equations (15) give rise to singular solutions of thefirst order. Eliminating z from (15), ~we obtain y' = A (x,y). (w 11). The singular solution, if any, of this equation, taken along with w3 ---- 0 will (in general) constitute singular solutions of the second order. If ws = 0 give on solution fox" z, z ---- 8 (x, y), then y' = A (x, y) = r Ix, y, 0 (x,y)]. The singular solution, if any exist, of this equation is given either b y bA ~A b ~ -- infinite, or by ~-~ = infinite. In most cases, w s = 0 will be an elementary variables.
function whose derivatives are finite for all finite values of the The infinities of ~-~ and ~-y can thus arise only
either (i) t h r o u g h the infinities of - ~ - ,
bY ' ~
~z
or (ii) on account of the equation bws - 0 Efor this may cause the bz derivatives of 0 (x, y) to become infinite].
Case (i). Let wl (x, y, z) -- 0 be the equation defined by - ~
or -@- or . ~ z
= infinite (wt being supposed to be distinct from the particular w3 t h a t we are considering). The condition of sufficiency t h a t wl = 0 ; w3 = 0 may constit u t e singular solutions of the second order for the given equations is t h e same as the condition t h a t wt (x, y, 8) = 0 may be a singular solution for t h e e q u a t i o n y ' = A (x, y).* This requires t h a t the equation y ' = ~bi (x, y, 0) should be satisfied b y the value of y ' given b y bw~ + ~w~ , + ~v,,~ (~O + b o y , ) = 0 in virtue of wl (x, y, 0) = O. * W e suppose that the particular w~ considered is not a function of ~ and v. If Ws be a function of u aucl v, the above condition is not sufficient to give singular solutions Of the second order. In fact, then, ws = 0; y ' = ~i do not give rise to 8hz.qular solutions at all, while wl = 0, Ws = 0 form singul~r solutions of the firs~ order.. (See ]~xample 12.) &2 I
680
C.N.
Srinivaslengar
This is equivalent to the condition ~
+ ~-
r
+
~
= 0
in virtue of w~ = 0. The equations wa = 0 ; y ' = ~b~ (x, y, z) . . . . .. (15) have been supposed to give singular solutions of the first order. They m a y be replaced by z = O (x, y) ; y ' -----*#l. But as the equations (15) and (16) are equivalent, it must be possible to derive the equation z' = ~b2 (x, y, z) f r o m (15). Therefore b9 + ~O ~ ~bt is equivalent to ~b2. Hence bw~ + bw~
~w~
= 0 in virtue of the equations wi ( x , y , z ) = 0 ; y ' = ~ b l . But, this is t h e condition t h a t the equations wl (x,y, z ) = 0 ; Y ' = r m a y give rise to a set of singular* solutions of the first order. Thus, in this case, there will be two different sets of singular solutions of the first order, one set given by wa = 0 ; y ' = r another given b y wl = 0; y ' = ~bl. The singular solution of the second order is given b y Wl - - 0, WZ =
0.
[Conversely, given the existence of two different systems of singular solutions of the first order given by w3 = 0 ; y ' ---- r and wl = 0 ; y ' = r we can usually infer the existence of a set of singular solutions of the second order, viz., wa = O, wl = O, but not always. bw3 ~b2 = 0, and
5w~
+
~w~
~bl +
bw~
The equations
5Wa
-~
5W~
+-~-~
r
~b2 = 0 are satisfied in v i r t u e
of the equations w3 = 0 and wi = 0 respectively 9 Hence the values of y ' and z' for the curve of intersection of w3 = 0 and wl = 0 satisfy the g i v e n differential equations. This is however provided t h a t the y ' and z' for this curve .are derivable from bw 3 bw~ y, bw3 z' +
and bwt ~ + ~~wt y
+ W
= o
, + bw~ ~ z' = 0 .
The exceptional cases are when the two surfaces w~ = 0 and w i = 0 touch each other throughout their curve of intersection (vide Example 7), or when the curve of ,ntersection is a double line on one of the surfaces.J Case (ii) The equation bws -- 0 arises as the equation in virtue of w h i c h 9
bZ
the solution z = 0 (x, y) obtained by solving wa =
0 m a y not be in t h e
* T h a t these s o l u t i o n s are singLflar does n o t n e c e s s a r i l y follow f r o m t h e p r o o f . All t h a t we c a n s a y f r o m t h e a r g u m e n t ~ is t h a t w~ = 0 ~ y" = ~ , c o n s t i t u t e s o l u t i o n e of s o m e kind. B u t if t h e s e s o l u t i o n s were p a r t i c u l a r , wx = 0 c a n n o t a s s o c i a t e i t s e l f w i t h a n o t h e r e q u a t i o n t o g i v e s i n g u l a r s o l u t i o n s of t h e seeand o r d e r .
681
S i n g u l a r Solutions off Di/ferential E.quations
s t a n d a r d form (continuous with continuous first derivatives). Now, instead of s t a r t i n g w i t h the equation (15), if we had started with (16), the except i o n a l case arises when ws = 0 cannot be solved for y in the prescribed form. The necessary condition therefore becomes either (iii) br
~r
br
infinite
(iv) 5wa __ 0. by The condition (iii) leads to a second set of singular solutions of the first or
order, precisely as in case (i)
9
Otherwise, w e h a v e w3=0"
'
bwa = 0 "
Ow 3
-- 0.
Again, any one of the three variables x, y, z may be taken as independent.
dy
eX we shall similarly
Hence, starting with the combination wa = 0; d-7 = ~ o b t a i n the condition ~wa -- 0. ~x
I n the absence of two different systems of singular solutions of the first order, we must have therefore wa = 0 ; bwa = 0 ; ~wa -- 0 ; bwa = 0, 5x by bz these being equivalent to two independent equations only. We thus arrive at the following t h e o r e m : T~IEORE~ II. Given the existence of one curve representing singular
solutions of the second order for the given &fferential equations, it follows that either there will be two different systems of singular solutions of the first order, or otherwise the c'urve is a double line on the surface ws (x, y, z) = O, which leads to the singular solutions of the first order. The surface w3 = 0 can be seen to form the focal surface or part of it, of the congruence of curves represented by the general solutions. W h e n there is only one set of singtdar solutions of the first order, wa = 0 forms t h e entire focal surface. When there are two sets of singular solutions of t h e first order, t h e focal surface breaks up into two proper surfaces. The condition in the first footnote in this section implies t h a t the focal surface w3 is n o t itself a surface of the congruence.
w I n connection with the above theorem, it will be proper here to refer to some results t h a t have been obtained from entirely different considerations b y A. C. Dixon in his paper (Ref. 2). Dixon confines himself to t h e Clairaut-equations y - x y ' = f ~ (y', z') ; z - x z ' =f2 (Y', z'). (1) The straight lines comprising the rectilinear congruence y - a x = fl (a, b) ; z - bx=f.2 (a, b) form bitangents to a surface, viz., the focal surface. T h e nodal curve (but not the cuspidal curve) of this surface will in general
682
C. N. Srinlvaslengar
furnish a singular solution of the second order for the differential equations of the congruence. (2) I n special cases, the focal surface of the above rectilinear congruence may become a developable. The edge of regression (which is known to be a locus of unodes) furnishes a singular solution of the second order. (3) A special case is considered wherein the Clairaut-equations represent the congruence comprising the inflexional tangents to a sudace. This falls as a particular case of (1) when the focal points that lie on any ray of the congruence coincide. Dixon proves t h a t if the surface contains a cuspidal curve (i.e. a locus of unodes), it furnishes a solution which, he says, i s of the second order, and t h a t the parabolic curve may also furnish a solution, if it happens to be a plane curve. I t may be mentioned t h a t in this case, t h e asymptotic lines of t h e surface form singular solutions of the first order.* It is well known t h a t the asymptotic lines do not in general possess an envelope, the exception being when the parabolic curve is a plane curve. When the parabolic curve is a plane curve it furnishes a solution of the differential equations. B u t this will not be a singular solution of the second order, for the parabolic curve is not a double line. I n fact, since the parabolic curve in t h e case considered is the envelope of a singly infinite system of inflexional tangents properly selected out of the congruence, it is included in t h e singular solutions of the first order. As regards the unodal locus, I do not agree with Prof. Dixon when he says t h a t it represents a singular solution of the second order. A singular solution of the second order must necessarily represent the envelope of t h e curves given by the singular solutions of the first order, and the curve in question does not possess this property. In fact, in w 4.5 of Dixon's paper, it is proved that the tangent to the unodal locus at any point is a hyperinflexional tangent thereat, i.e., a tangent having contact of the third order. Now, when the equations of the congruence of the inflexional tangents are written down, any system or systems of hyper-infiexional tangents are * Th e a s y m p t o t i c lines m a y n o t p r o b a b l y e x h a u s t , in every case, all t h e singular solutions of the first order. W h e n a surface is o f degree higher t h a n four, it m a y h a p p e n t h a t a s t r a i g h t line m a y be an inflexional t a n g e n t at one point P on t he surface, and an o r d i n a r y t a n g e n t at a n o t h e r point Q. W e might possibly h a v e therefore a curve on t h e surface every t a n g e n t to which touches t h e sucface inflexionally elsewhere. Such a c u r v e e~idently furnishes a singular solution of the first order (we m u s t r e m e m b e r t h a t t h e inflexiona] ta.ngents satisfy t h e differential equations considered all along t h e i r l engt h, and n o t me.rely at tt,e points where t h e y t o u c h the surface inflexionally), b u t is n o t an a s y m p t o t i c line.
Singular Solutions of Differentia/Equations
683
usually included in them. Hence the unodal locus is the envelope of a particular o=1 of inflexional tangents included in the congruence. It provides therefore a singular solution of the first order only. Consider Dixon's E x a m p l e :
EXAMPLE 4. The system of lines
y = 3a2 b*x + 89 (1 - 6 a 3) Z = b 3 (1 2 7 . a9) 3 x - - ~a~bS forms one system of infiexional tangents on a certain surface of degree 160 twelve. The cuspidal curve is given bv. x = 8 t~ ; y -- 3 t 9"' Z = 4 0 t s. This curve will be found to be the envelope of the ool lines obtained by p u t t i n g 2a3 = 1 in the above equations, and these lines are the hyperinflexional tangents at the unodes. The curve is thus included amongst the singular solutions of the .first order.
[w 76]. Summary of Results. w 16. THEOREM 3. Let the given differential equations be reduced to the form y ' = ~bI ( x , y , z ) , z ' = r (x,y,z), and let their general solutions be written u (x, y , z ) = a ; v (x, y , z ) = b . Let w (x, y , z ) = O denote an equation in virtue of which a f r s t partial derivative of r
or r
(or ~ or ~-~.) as u,ell as
a first partial derivative of u or v becomes infinite. The singular solutions (if any exist) of the given equations will be included in the equatibns w=O; y'=r orz'=r Ta~OR~M 4. I f when any of thefirst partial derivatives of u becomes infinite, none of the first partial derivatives of v becomes infinite, and vice versa, we must search for singular solutions of the first order from the following equations : ~u 3u ~u ~-- or ~ or ~ = infinite; v (x, y, z) = b . . . . (i) by ~v by ~---~or ~ or ~ ---- infinite; u (x, y, z) = a
. . . .
(ii)
Let the fi,nctions defined by the first halves in (i) and (ii) be called w3t = 0 and w~e = O. Then the singular solutions of the second order (if any exist) will be given
by
wal =0; o r b y zv31 =
w~z=0 O" ~w~1 = '
b~
o r b y w32 ---- 0" ~wa~ 9
~
. O"
~w~, _
'
=
.
.
.
.
0 9 ~w~
_
b~]
bW~2 __
0" '
by
'
'
bz
.
.
0
~z
bWs2
0"
.
(iii)
.
..
(iv)
.
..
(v)
"
-- 0 "
684
C. N. S r i n i v a s i e n g a r
[w 77].
Generalisation.
w 17. The methods followed in the previous pages are obviously capable of being extended to equations involving more variables. Only one theorem will be written down here. THEORE~I 5; Let the general solutions of the system of equations y / = T r (x, Yl, Y2, 9 9 ", Y,,) be written u,. (x, y~, . . . , y , , ) = a,., (r = 1, 2, . . . , n). Let w,. ---- 0 denote the eq~.t.atio~ in virtue o / w h i c h any of the first partial derivatives of ur becomes i~finite. I f each of the w's possesses this property with respect to one of the u's only, the singular solutions of the first order must be sought for from systems of equations containing one of the w's and its non-corresponding ( n - 1 ) r ." the singular solutions of the second order may be obtained from systems of equations containing two of the w's and their non-corresponding ( n - 2) u's, and so on ; singular solutions of the n t]' order may be given by al~ the w's tahen together, if these are n in number. This process m a y not e x h a u s t all t h e singtflar solutions of the second and higher orders, t h o u g h those of the first order are obtained exhaustively. I shall not discuss this in detail here.
[w 18]. w 18.
Examples.
ExAxP~E 5.
y = xy' + / 2 + z' ; z = x z ' + y ' z ' The singular solutions of the first order, as o b t a i n e d from the t h e o r e m quoted in w 10, are given by w = { 27 ( z - x y ) - 2 x ( x 2 - 9 y ) } 2 - 4 (x~+3y) s = 0 } .. .. (17) associated with y = -- 88(x + k) 2 + k 2 whale the singular solutions of t h e second order are given b y the double line o f w = 0 , viz.,x 2 + 3 y = 0 ; .z~ + 2 7 z = 0 . The following remarks as regards the function w will be interesting. The general solutions of the given equations, viz., y = a x + a 2 + b ; z = b x + ab are given as the partial intersections of the surfaces a(x§ . . . . . . . . . . (18) b3--b~y--bzx+z ~ = 0 . . . . . . . . . . (19) Instead of solving for a and b, and finding when the derivatives of t h e functions u ( x , y , z), v (x,y,z) become infinite, we can do t h e equivalent process of finding the a- and b-discriminants of t h e equations ( 1 8 ) a n d (19). It wilt be found that the discrimi'nants are identically the same, b o t h simplifying to the relation w = 0. This is n o t a mere accident, b u t is generally true whenever we are dealing with equations which are wholly algebraic and rational. I t m a y be proved as follows : - - L e t a and b be
Singular
685
S o h l t i o n s o f D i / f e r e ~ t i a l &'qualio~zs
eliminated i n t u r n from the equations f r (x, y, z, a, b)-~ 0, (r-----1, 2), a n d let the eliminants be written after rationalisation in the form u (x,y, z, a) -- O, v (x,y, z, b) ---- 0. The a-discriminant of u (x, y, z, a) ----- 0 represents t h e condition t h a t the solution for a MIGHa: give a function one or more of whose first partial derivatives becomes infinite.* A similar statement holds for t h e b-discriminant of v---- 0. If the result of eliminating a and b from fl ---- 0 ; f2=0
'
b ( f l ' f - ' ) _ 0 be written w ( x , y , z ) = 0 , b (a, b)
then w----0
denotes
the
condition fil virtue of which the solution for a or for b, or for both, from the equations f l = 0; f2---- 0 might be a function one of whose first p a r t i a l derivatives is infinite. Hence either of the following possibilities must be satisfied: (a) The a-discriminant of u, and the b-discriminant of v must b o t h be t h e same as w. (b) Otherwise w can be broken up into two factors, viz., the a-disc r i m i n a n t of u and the b-discriminant of v.~ The condition (b) can naturally be satisfied only in special cases. (An instance is provided by ~ x a m p l e 7.) When w = 0 is non-degenerate, as in the present example (originally given by Serret), condition (a) m u s t necessarily be satisfied, and there can exist only one set of singular solutions of t h e first order. The singular solutions cannot be obtained by using Theorem 4. h a v e to be obtained by using Theorem 3.
They
[ F o r s y t h (Ref. 5) writes the singular solutions of the first order in t h e form y =
-
88 (x +
k)
+
k2; z =
-
. . . .
(20)
- - a form simpler t h a n t h a t in (17). In fact, substituting y = --88 (x + k) -~ + k 2 i n w - - - - 0 and solving for z, we obtain two values one of which is z = --~ k ( x - - k ) 2. The curves (17) are thus equivalent to the curves (20) together with the curves y ------[ (x+k)2-t-k2; 108 z ~- 27k s - 9 k x 9- - 2x'~ .. (21) * T h i s is u n d e r t h e s u p p o s i t i o n t h a t t h e s y s t e m of s u r f a c e s u = 0 does. n o t p o s s e s s a s u r f a c e - l o c u s of d o u b l e p o i n t s . I t is h o w e v e r a k n o w n f a c t t h a t if u = 0, o r v = 0 p o s s e s s e s s u c h a s u r f a c e - l o c u s of d o u b l e p o i n t s , t h i s s u r f a c e is i n c l u d e d as a n e x t r a n e o u s f a c t o r i n t h e f o c a l s u r f a c e w = 0 of t i l e c o n g r u e n c e f~ = 0 , .f2 = 0. W e d o n o t , . a g a i n , t a k e i n t o c o n s i d e r a t i o n a n y p a r t . i c u l a r s u r f a c e of t h e s y s t e m u = O t h a t m i g h t a p p e a r i n t h e a-discriminant. t I n b o t h c a s e s , i t is c l e a r t h a t t h e f o c a l s u r f a c e of t h e g i v e n c o n g r u e n c e f l = 0 , f2 = 0 is i d e n t i c a l w i t h t h a t of t h e d e r i v e d c o n g r u e n c e u = 0, v = 0, a l t h o u g h t h e " s u r f a c e s of t h e t w o c o n g r u e n c e s " a r e diff.erent.
686
C. N. S r i n i v a s i e n g a r
The curves (21) are not at all solutions of the differential equations considered. Thus, when we take the three equations r = 0" r '
= 0" ~ (~1, ~2) = 0 (w 10) ' ~ (y', z')
only, omitting the two other equations in the theorem quoted in w 10, we obtain, in the present example, a singly infinite number of extraneous curves in addition to the singular solutions. The congruence does not possess any multiple points or tac-points. The occurrence of the extraneous system in spite of this fact is a noticeable thing. The extraneous curves are those that satisfy (15) but not (16).] EXA_M__PLE6. To f i n d the focal surface of the congruence y = ax + a 2 + ab + b ; z =
bx + ab + b2.
This example is taken here merely to illustrate the use of the remarks made in the previous example. The usual method, viz., to eliminate a and b between the given equations and their Jacobian w.r.t, a and b may be replaced by the simpler but equivalent method of eliminating, say a between the given equations, and then finding the b-discriminant of the resulting equation.
EXAMPLE
7.
The congruence y = ax + a2 + ab + b ; z = bx + b% The focal surface w = 0 breaks up into two distinct surfaces wa=-4x s + 1 6 x y + 4xO-y + 16ye + 8yz + z~ + 1 2 x z - - 1 6 z = 0 and wb=-x 2 + 4z = O. Eliminating b, the congruence m a y be considered as included in z (a + 1)2 = ( y - a x - - a 2) (x + y - a s) z = b x + b 2. The differential equations y = xy' + / 2 + y ' z ' + z' ; z = xz' + z'~ admit of two sets of singular solutions of the first order. One set is included in the equations w= = 0 ; z = bx + b 2. The actual equations are y = -- 88 (x + b) 2 + b ; z = bx + be. We might expect, the other set to be included in w a = 0; z ( a + l ) ~ = ( y - - a x - - a 2) (x+y--a'-'). B u t this is not so. The surface wa = 0 is found to be the common envelope of the two systems z = b x + b ~, and z ( a + l ) - ~ ( y - - a x - - a ~.) ( x + y - - a ~ ) . The second set of singular solutions of the first order has therefore to be found out by actually solving the equations ~g
x'- + 4z = 0; y = { x y ' + y'2 -
~.
w , = 0 does not possess a double line, and since the equation y = 89xy' + y ' ~ - 2 does not possess a singular solution, the given equations have no singular
Singr & r S o l u t i o n s o f 29ifferen!ial E q u a t i o n s
687
solutions of the second order. The reason why wa = 0; wb = ' 0 do not constitute solutions is that they touch each other along the entire curve x2+8x+16y=0; x 2 + 4 z = 0. (Vide w [These geometrical peculiarities are found to be generally true for any congruence of the type y = ax + A (a, b) . . . . . . (22) z =bx+f.2(b) . . . . . . (23) There will be two distinct focal surfaces, one of which is the envelope of the system (23). I,et (23) be solved for b in the form v(x, y, z) = b. Eliminating b between (22) and (23), we have y - ax = f l (a, v) . . . . . . (24) I t now follows* that the envelope of the system (23), say El is also all envelope of the system (24). The system (24) possesses another envelope, say E2, which will constitute the other focal surface of the congruence. Now E1 touches the surface y - - a x = f l (a, v) for any given value of a, all along the curve of intersection; so does E2. Hence, at the point of intersection of the three surfaces ~i, t~_~, and y - a x = fl (a, v), they have a common tangent plane. Therefore, the two focal surfaces of the congruence (22), (23) touch each other all along their curve of intersevtion.~ Ex.~--~PL~: 8. dx dy 3 + 2(z + x + 2y)89 = '2(z--x--y) 89 1 + 2(z + x + 2y) 8 9
dz y)89 4 (z--x--y) 89 (z + x + 2y)89 (~. J. ~..I-Ell.)
The general solutions are u=x+(z-x-y) 89 v-y+(z+x+2y) 89 There are two sets of singular solutions of the first order, viz., z = x + y ; v = b , and z + x + 2y = 0 ; u = a. The singular solutions of the second order are given by z = x + y ; z + x + 2 y = O . EXAMPLE 9.
Consider the equations of :Example 2. Theorem 4 cannot be used, since z = x + y makes the first derivatives of both u and v infinite. Theorem 3 readily gives the singular solutions of the first order in the form z = x + y ; x2--.y2=C.
In this example, however, it is possible to find a function 8 (u, v) which is free from irrationals. If we take the general solutions as included in u = a ; v' -- u2--v 0" = (x2--y 9.) ( 1 - z - x - y ) = c, * Vide Toholvu Mathematical Journal, 39, 90-91.
C. N. Srinivasiengar
688 the singular solutions z = z + y , v'
can be obtained by Theorem 4 as included in ~,XAMPLE 10.
Consider the differential equations having for their general solutions u=x+y[log(z-x-y)]-l= a v ~ y + x [ log ( z - x - y ) ] - ~= b (M. 5. ~ . Hill.) The singular solutions of t h e first o~der, viz., z --. x + y ; x ~ - y 2 = c m a y be obtained by using Theorem 3. I n this case, 0 (u, v) cannot be found so
as to be free from the transcendental expression. EXAMPLE II. In example 3, we obtain the solutions z - - x - y ----0, x = constant, by using Theorem 3. W e have seen that these are not singular solutions. They form the characteristic curves on the envelope of either of the systems u=a, a n d v = b. (Seew
:EXA.I~I:'LE 12. Y'_J + Y- -'2x --(y'+ 3 2
yt2 1)y-z=0;
y' + ~ -
+z'=0.
This example, given by Mayer (Ref. 4) has the peculiarity t h a t it possesses
a single singular solution of the first order,
viz.,
y
=
4
'
z
=
4
24"
The reason is t h a t the focal surface of the congruence defined b y the general primitive happens to be a p a r t i c u l a r surface of the congruence 9 'The Singular solution represents a double line on the focal surface. The double line would have g i v e n a singular solution of the second order, but for the a b o v e peculiarity about t h e focal surface. [w16779-2,$].
T h e Geometry o f the S i n g u l a r S o l u t i o n s .
w 19. Let us first consider t h e case where the two systems u -~ a a n d v = b do not possess a c o m m o n enveiope. A necessary condition t h a t the surface w ( x , y , z ) = 0 should be an envelope of the system u ( x , y , z ) = a is t h a t one of the first p a r t i a l derivatives of u becomes infinite when w = 0. I n fact, this c o n d i t i o n is in most cases sufficient.* We shall however precisely write down the 'conditions of s~fficiency in t h e form bw ~u ~w ~)U Lt ~y ~-x__= Lt ~x__ and ~y (.25) w--> 0 ~ ~w w -.-> 0 5u 5w 5z 5z bz ~z * Vide Tohokl, Math. Journal, Vol. 39.
689
S i n g u l a r Solutions o f Diff'ereulial E q u a t i o n s Sufficient conditions t h a t w = 0 ; v = b may given differential equations are that
proxdde solutions of t h e
lu, v)) ~ (z, x) _ I,t ~(w,v! w--> 0
~ (z-~ iu, v)~
o (y, z)
and
(y,z)
5 (x, Y) I,t ~ (w, ;) = w--> 0
5 (x, y)~
(26)
(y, z)
The t w o sets of conditions (25) and (26) are equivalent.
Thus, (u, v) ~v Lt 5 (z, x) ~x w - + 0 ~u~v -- by bz bz 5z
~u ~ (w, v) Lt ~x ~(z, x) , using (25). w - + 0 ~"~ = bw~v 5z ~z 5z b (u, v) ~ (w, v) Similarly, (y, z) _ ~ (y, z) Lt buov 5w by w->O bz 3z bz 5z Hence, b y division, we obtain one of the conditions in (26). The other condition is similarly obtained. The algebra m a y be reversed with slight manipulations. The conditions of sufficiency t h a t w---0 might envelope the s y s t e m u - - a are t h u s t h e same as the conditions of sufficiency t h a t w = 0 ; v = b m i g h t be solutions of the given differential equations, w being s~s to be not a function of u and v. W h e n t h e singular solutions of the given differential equations are obtainable from Theorem 4, i.e., when the two systems u = a and v = b do not h a v e a common envelope, we h a v e therefore the following theorem : THEOREM 6. The singular solutions of the firs~ order will be given either as the curves of intersections of the surfaces v = b with the envelope of the system u = a , or as the curves of intersection of the surfaces u = a with the envelope of the system v=b. The intersection of the two envelopes usually provides a singular solution of t h e second order. (Vide w 14 for exceptions.) We have assumed t h a t the envelopes are not included in the surfaces of the congruence. w 20. The following property of the focal surface will be discussed in a separate note elsewhere. For the congruence F , ( x , y , z , a, b ) = 0, (r = 1,2), there exists in general a ~ 1 surfaces of the congruence which do not necessarily t o u c h t h e focal surface. These surfaces are determined by the e q u a t i o n (a, b, ~
F
= 0, which js the result of eliminating x, y, z from
=0.F2=0.L 9 '
ba
+
~F I ~ 5b
da
=0' '
5F~ ~a
+
bF2
d~
bb
da
--0.
690
C. N . S r i n i v a s i e n g a r
For the congruence u = a, v = b where the two systems u = a and v = b do not possess a common envelope, the surfaces v = b are to be regarded as surfaces of the congruence which do not t o u c h the focal surface defined b y the envelope of u=a, except p e r h a p s for special values of b. If, however, the systems u==a and v=b possess a common envelope, this envelope touches in general (excepting perhaps a finite number of surfaces) all surfaces of the congruence. For the case considered theorem 9
in w 19,
we h a v e therefore the following
THEOREM 7. The singular solutions of the first order are the curves of intersection of the Jbcal surface with those surfaces of the congruence which do not touch the focal surface. W h e n there are two or more focal surfaces, each of these must be considered in turn.
w 21. I, et us now start w i t h a given congruence of curves F,. (x, y, z, a, b) = 0, (r = 1, 2) where t h e F ' s are algebraic a n d rational, and consider t h e problem of obtaining directly t h e singu]ar solutions of the corresponding differential equations. W e c a n n o t reduce this problem so as to be able to use Theorem 4, for if F1 = 0 a n d F2 = 0 are p u t in t h e form u-----a and v = b, we have seen while discussing E x a m p l e 5, t h a t as a rule, the two systems u = a and v = b will h a v e a common envelope. The following will be a geometrical m e t h o d from first principles" L e t P be a n y point on t h e curve 6f i n t e r s e c t i o n of t h e surface F1 [(x, y, z, a, r (a)] = 0 for an assigned value of a w i t h t h e surface El enveloping this system, [b = r being a pre-assigned relation between b and a]. These two surfaces will h a v e t h e same normal PN1 at P. If P also h a p p e n s to lie on the curve of intersection of the surface F=[x,y,z,a,r (a)] = 0 for the same value of a w i t h t h e envelope E2 of this system, and if PN2 be the common normal at P, t h e n PN1 and PN= m a y be regarded as n o r m a l s either to E1 and E.~, or to t h e surfaces Fx = 0, F= = 0. B u t the intersection of the latter surfaces is a particular curve of the congruence, and satisfies t h e differential equations of the congruence. H e n c e t h e curve (El, E2) also satisfies these equations at P. The point P being common to four surfaces should satisfy the four equations Fl(z, y, z, a, b) = 0 ; ~F13a + 3Flb__bda =0db F2(x,y, z, a, b) = 0"' ~F~ ~a + ~Fo ~ b ' ddb a =0
..
(27)
The curve (F~l, E~) is the same as the locus of P. E l i m i n a t i n g x, y a n d z from (27), we obtain a relation 8
a, b, ~-~
= O. The relation b = r (a)
S i n g M a r Solutions o f D i fl'erenlial ~Equettlons
691
should be taken so as to satisfy this equation. Let the general solution of 8 = 0 be b = % (a, k) where k is an arbitrary constant. Substituting and eliminating a, we obtain the equations of the curve (El, E2). The equations involve an arbitrary constant k, and give the singular solutions of the first order. Extraneous loci may however also occur. If 0 = 0 admits of a singular solution, this will lead to the singular solutions of the second order. The equations (27) have been deduced from analytical considerations in F o r s y t h (Ref. 5) and in Dixon (Ref. 2). Another method is given in the next article. w 22. The four equations (27) are the same as those t h a t determine the surfaces of the c o n ~ u e n c e which may not touch the focal surface. (w 20.) This proves t h a t Theorem 7 continues to hold good, in the main, for congruences of the type considered in w 21. There is another theorem which holds good for both the cases of w167 19 and 21, viz. :
THEOREiVs 8. We can select singly infinite systems of curves belonging to t h e congruence, so as to possess an enveloping curve. Any such enveloping curve will give a singular (in general) solution o/ the first order. I / the curves representing singular solutions of the first order possess an envelope of their own, this envelope will represent in general a singular solution of the second order. This t h e o r e m is well known and will be found mentioned in the works of Goursat and Dixon. When applied to the congruence of w 19, it gives an a l t e r n a t e proof of Theorem 4, and when applied to the congruence of w21, it at once gives the equations (27). w 23. Although Theorems 7 and 8 hold good for the congruences of w167 19 and 21, they do not remain true for the congruence u=a, v=b wherein the two systems u = a and v= b possess a common envelope. I n this case, the so-called focal points on any curve, i.e., the points of intersection of tile three surfaces u=a, v=b, and the focal surface are points where the two surfaces u = a and v = b t o u c h each other. They are therefore double points on .the curve. Thus in E x a m p l e s 9 and 10, the singular solutions z = x + y ; xg--y~ c are t o be regarded as the loci of double points of singly infinite systems of curves of t h e congruence, rather t h a n as their envelopes. [The congruence Fr (x, y, z, a, b) = 0, r = 1, 2 can in general be considered as forming part of the congruence u = a, v = b (obtained by solving for a and b) where the systems u = a and v = b have a common envelope. W h a t is s t a t e d here is t h a t if the general solutions of the differential equations are completely represented by u = a, v = b where the two systems of surfaces
692
C. N . S t i n i v a s i e n g a r
h a v e the same envelope, t h e n Theorem 7 a n d t h e first part of Theorem 8 a r e not true.] w 24. Theorem 2 t a k e n w i t h t h e second p a r t of Theorem 8 gives t h e following geometrical p r o p e r t y : I f the congruence F, (x, y, z, a, b) = 0, (r = 1, 2) possess a non-degenerate
focal surface, and i f the envelopes o/ singly infinite systems of curves selected out of the congruence possess an envelope of their own, the latter will be a double line on the focal surface. [w167 25-26]. The Nodal Locus.
w 25.
I t was mentioned in w 21 t h a t extraneous loci m a y sometimes occur; which do not c o n s t i t u t e singular solutions. We shall not a t t e m p t here a s t u d y into t h e n a t u r e of t h e possible extraneous loci. F o r s y t h seems to believe t h a t t h e nodal locus is always one such factor, for he writes (Ref. 5, p. 180), " t h e equation I =
(F~, F2)
~ (a, b) -- 0 is satisfied at every node. "
This is not always true. If we consider a non-degenerate skew curve which is t h e complete intersection of two proper surfaces, there are two w a y s in which a node can arise. A d o u b l e point on either surface is also a double point on the curve of intersection. A second t y p e of node is defined by t h e theorem : " If two surfaces touch, the point Of c o n t a c t is a double point on t h e i r curve of intersection."* If now we consider the congruence, F~(x, y, z, a, b ) = 0, ( r = 1, 2), it is easy to see, b y constructing examples, t h a t a node of either of the types on a n y curve of t h e c o n g r u e n c e does n o t necessarily satisfy ~ (F1, F.o) _ 0. (a, b) w26. The following example (Ref. 5, Art. illustrates a very different aspect.
208) is instructive a n d
Ex~PL~
13. Y l - x ~ + y2+ z 2 - a 2 = 0 ; F~=-ax2+ by2 + 2abxy--a 3 = O. The equations (27) give the extraneous locus y = 0, z = 0. This is a nodal locus of t h e curves, b u t not of either of t h e t y p e s described above. W h e n we p u t b = 0, t h e curves of the congruence reduce t o x 2 - a 2 = 0 ; y~ + z .2 = 0. T h e curve x = a ; y2 + z2 = 0 consists of two lines meeting at .(a, 0, 0). The.line y = 0, z ----0is t h u s obtained as the locus of the nodes of degenerate curves o f the system. The appearance., of t h i s t y p e of a nodal locus as a n - e x t r a n e o u s , factor while dealing with t h e equations (27) requires explanation. My e x p l a n a t i o n is t h a t its occurrence m u s t be accounted for b y t h e f a c t t h a t the line y = 0, z = 0 meets e v e r y curve of t h e congruence, * Salmon, 2tnaly~ica~Geom'etril of Three D~men~on~, ~ .203.
S i n g u l a r Solulions
o/"DifferenHal Equations
693
T h r o u g h a n y point in space there pass in general a finite number of curves of the congruence Fl = 0, Fe -- 0. If, however, (xl, Yl, zl) be a point t h r o u g h which pass an infinite number of curves of the congruence, then the equations Yl = 0 , :F,2= 0 cannot give definite solutions for a and b when x----xl y = y , , z = z~. Hence, by the theory of implicit functions, we must have (F~,F2) -~(a, b) ----0 at (xt,Yl, zl). At such a point, the four equations (27) reduce
to three only. If there be a locus of such points, this locus will figure in the process described in w 21.* A curve which meets every curve of the congruence constitutes as a rule a singular solution of the first order for ~he differential equations of the congruence. For through every point on the curve there pass a ool curves of the congruence, and there will usually be one curve of the congruence which touches the given curve at the point. Hence the given curve is the envelope of a properly choser~ ool of curves of the congruence. I t provides therefore a singular solution of the first order. E x a m p l e 13 presents an exception to this general statement. Consider any point (a, 0, 0). The surface FI = 0 is independent of b and the line y = 0 , z = 0 d o e s not happen to lie in the tangent plane t o F 1 = 0 a t (a, 0, 0). I t is therefore impossible to find out the value of b such t h a t the corresponding curve of the congruence has y = 0, z = 0 as the tangent at
(a, o, o). REFERENCES. 1.
1%~:.ft. M. H i l l
..
..
.Proc. Z o n d . M a t h . Soc. (2) 16~
2.
A.C. Dixon
..
..
P h i l . T r a n s . Roll. Sor London, 1895, P a r t I A .
3.
E. Goursat
..
..
A m e r i c a n J o z , rnal o f M a t h e m a t i c s , 11. ( S e c t i o n I I o f the paper.)
4.
/k. M a y e r
..
9.
M a t h . A n n a l e n , 2Z.
5.
2~. R . F o r s y t h
..
..
T h e o r y o f D$fferengial EquaMons, 3, w167 197-208.
6.
Hamburger
..
..
CrHle's J o ~ r n a l , B d . 122.
7.
Serret-Scheffers
..
..
Z e h r b u c h clef D~fferent{=l uncl integral-rechnz~ng, B d . 3.
8.
C.N.
9.
Srinivasiengar
Do.
..
..
..
Toh'o~i M a t h . J o u r n a l , 30. I~id.,
39.
* T h i s is a p p r o x i m a t e l y e q u i v a l e n ~ t o t h e f o l l o w i n g w e l l - k n o w n r e s u l t : - - ' " I f a g i v e n c u r v e m e e t s e v e r y ~ u r v e of a c o n g r u e n c e , t21e g i v e n c u r v e lies o n t h e f o c a l s u r f a c e . " T h ~ t h e o r e m i8 h o w e v e r n o t ~ e c e s s a r i l y ~rae, u n l e s s t h e f o c a l s u r f a c e is b l i n d l y d e f i n e d so
(F,,
F2)
a s t o i n c l u d e a n y s u r f a c e o b t : ~ i n e d b y e l i m i n a t i n g a a n d b f r o m F I = 0 . . F2 = . 0 , - ~ (a, b )
= 0, i r r e s p e c t i v e of a n y g e o m e t r i c a l p r o p e r ~ i e s t h a t t h e ~resulting s u r f a c e m a y o r m a y n o t p o s s e s s . F o r e x a m p l e , t h e c u r v e r~ = 0, r z = 0 m e e t s e v e r y c a r v e of t h e c o n g r u e n c e u - - r-A ----a ; v = b w h e r e r , a n d r~ a r e r a t i o n a l i n t e g r a l f u n c t i o n s oDx, M, z . T h e e n v e l o p e of t h e s y s t e m v = b c o n s t i t u t e s t h e s o l e f o c a l s u r f a c e , b u t c o n t a i n t h e c u r v e r~ = 0, r~ = 0.
this does not
necessarily
