A class of nonlinear differential equations with exact solutions

CoA R E P O R T AERO. NO. 187

« * » * ^ * ^^[^\ ^ - - ^ Ü g s S ^

THE COLLEGE OF A E R O N A U T I C S

C R A N F I E L D

A CLASS OF NONLINEAR D I F F E R E N T I A L EQUATIONS

WITH EXACT SOLUTIONS

by

CoA REPORT AERO. NO. 187 November, 1965

'THE

COLLEGE OF AERONAUTICS

CRANFIELD

A Class of Nonlinear Differential Equations with Exact Solutions

b y

-P . A . T . Christopher, D . C . A e . , A . F . I . M . A .

SUMMARY

In Ref. 1, Smith has obtained exact solutions to a class of nonlinear differential equations of second order which are closely similar in form to equations arising in physical applications. The present paper extends these results to a wider class In which the equation may be of any order and in which there is a greater freedom of choice of the coefficients. As an application these results are used to test certain stability criteria arising from the use of

CONTENTS

Page Summary

1. Introduction 1 2. Synthesis and solution of a class of differential equations 1

3. An application 5 4. Equations of higher order 8

1

-1. Introduction

T h e c l a s s of differential equations defined by

x " + f(x).x' + g(x) = 0, E q . 1.1 w h e r e x' = d x / d t and f(x), g(x) a r e t e r m i n a t e d power s e r i e s , i s of c o n s i d e r a b l e i n t e r e s t t o e n g i n e e r s , and o t h e r s , s i n c e it d e s c r i b e s phenomena which a r i s e in a wide field of p h y s i c a l s i t u a t i o n s . See for e x a m p l e R e f s . 2 and 3 . It i s unfortunate, for t h o s e s e e k i n g to utilize E q . 1.1 in a p p l i c a t i o n s , that the g e n e r a l solution i s not known. T h e solutions t o a v e r y r e s t r i c t e d s u b - c l a s s of E q . 1.1 a r e known and will be found in Ref. 4, pp. 542-594. M o r e r e c e n t l y Smith, in Ref. 1, h a s obtained a solution t o a v e r y useful s u b - c l a s s of E q . 1.1 defined by

f(x) = (n + 2)bxn - 2a, g{x) = x [c + (bx" - a ) ] , E q . 1.2 w h e r e a, b , c, and n a r e c o n s t a n t s . T h e p r e s e n t p a p e r s e t s out t o extend t h e s e r e s u l t s to a wider s u b - c l a s s and to d e m o n s t r a t e t h e i r u s e in t e s t i n g s t a b i l i t y c r i t e r i a .

2. S y n t h e s i s and solution of a c l a s s of differential equations The s t a r t i n g point in t h e s y n t h e s i s i s the f i r s t o r d e r equation

X' + x.A(x) - U ' ^ U ' x = 0, E q . 2 . 1 w h e r e x and U a r e functions of t . R e - w r i t i n g E q . 2 . 1 in the f o r m

U " ' U ' = x - ' x ' + A(x) E q . 2 . 2 and differentiating with r e s p e c t to t gives

d ( U ' ' u ' ) / d t = U " ' U " - ( U ' ' u ' ) ^

= x " ^ x " - ( x ' ^ x ' ) " + x ' . d A / d x Adding t o t h i s the s q u a r e of E q . 2. 2 gives

d ( U " ^ u ' ) / d t + ( U ' ' u ' ) ' ' = U " ' U "

= x ' ' x " + X ' . d A / d x + 2x~^x'A(x) + A*(x) = x " ^ [ x " + x'(2A(x) + x . d A / d x ) + xA^(x) J

Using t h e s e r e s u l t s for U "^U ' and U "^U", t h e r e m a y b e f o r m e d the equation U ~' [ u " + ^ ( t ) . U ' + ^(t). U J = x "^ [ x " + x'(2A(x) + X. dA/dx)

2 -w h e r e F ( x , t ) = f(x) + ^ t ) , E q . 2 . 3 G(x,t) = x|_A*(x) + m ) . A ( x ) + ^ t ) j E q . 2 . 4 and f(x) = 2A(x) + x. d A / d x E q . 2 . 5 It follows that if U(t) i s f i r s t c h o s e n t o s a t i s f y

V"+'i>{i).V' + Ht).V = 0, E q . 2 . 6

then any solution x(t) of E q . 2 . 1 will satisfy

x " + F ( x , t ) . x ' + G(x,t) = 0. E q . 2 . 7 F r o m the f o r m of F ( x , t ) and G ( x , t ) it i s c l e a r that E q . 2. 7 defines an e x t e n s i v e c l a s s of differential equations for which ^(t), ir{t) and A(x) m a y be c h o s e n . T h e choice of t h e s e functions i s l i m i t e d by the need t o k e e p E q . 2 . 1 in a r e a d i l y i n t e g r a b l e f o r m . In fact it h a s been i m p l i c i t in the s y n t h e s i s of E q . 2 . 7 t h a t the g e n e r a t i n g E q . 2 . 1 s h a l l have B e r n o u l l i ' s f o r m , and n e c e s s a r i l y r e s t r i c t s A(x) to the f o r m

A(x) = b a ( t ) . x " -a/S(t) E q . 2 . 8 It i s d e s i r e d to r e s t r i c t attention to the s u b - c l a s s of E q . 2. 7 for which F and G

a r e explicit functions of x only. In o r d e r t o do t h i s the functions a ( t ) e t c . will be chosen a s follows:

a ( t ) = /?(t) = l , <t>W=k and i^(t) = c, w h e r e k and c a r e r e a l c o n s t a n t s .

On the b a s i s of the p r e v i o u s a s s u m p t i o n s E q . 2. 6 b e c o m e s U " + kU + cU = 0,

which h a s the g e n e r a l solution

3

-C a s e 3 . ff z e r o , X = P , r e p e a t e d root C a s e 4 . p z e r o , X = ± iff

T a k e f i r s t C a s e 1; then

U = e (A e + B e )

= e " * A(Cosfft + i Sinfft) + B(Cos ff t - i Sin fft)

Since A and B a r e a r b i t r a r y c o n s t a n t s then they m a y be chosen t o be complex such that A = C + i D and B = C - i D, giving 2C Cos fft - 2D Sinfft

U = e ' ' ^ [ 2 C

which m a y be w r i t t e n a s U = e ' ' ^ P ( t ) , E q . 2 . 1 0 w h e r e P ( t ) = E Sino-t + F Coso-t E q . 2.11 and E , F a r e a r b i t r a r y c o n s t a n t s . In C a s e 2, U = A e ' ^ i * + B e ^ 2 * , E q . 2.12 w h e r e X , X a r e r e a l . 1 2 In C a s e 3, X, = \ = p and A » i s no l o n g e r l i n e a r l y independent of B " ; thus E q . 2. 9 i s no l o n g e r the g e n e r a l s o l u t i o n . T h i s c a s e will be d i s r e g a r d e d . In C a s e 4 , p = o, thus

U = P ( t ) E q . 2 . 1 3 and r e s u l t s for t h i s c a s e m a y r e a d i l y be d e r i v e d from C a s e 1.

With the a s s u m p t i o n s m a d e above A(x) = bx" - a, giving f(x) = 2(bx" - a) + nbx" = (2 + n) bx" - 2a,

F ( x ) = (2 f n ) b x " + (k - 2a),

G(x) = x [ (bx" - a)* + k(bx" - a) + c J

4

-and Eq. 2. 7 reduces to the sub-class defined by

x " + ["(2 + n ) b x " + (k - 2a)1 . x' + b«x ^" "^ ^+b(k - 2 a ) x " "^ ^

+ (a* + C - Ka)x = 0 Eq. 2.14 The class of equations discussed in Ref. 1 is obtained by putting k = 0 in

Eq. 2.14. The advantage of the present form lies in the greater freedom of choice of the relative ranges of the coefficients, and, thereby, more generality.

Solutions to Eq. 2.14 will now be obtained by substituting the appropriate expressions for A(x) and U "'U' in Eq. 2.1 and integrating. Substituting for A(x) gives

X' +x(bx" - a) - U ' ' U ' x = 0

or

X' - F a H - U - ' U ' 1 X = -bx ""^ ^ Eq. 2.15 This is in Bernoulli's form and may be reduced to linear form by multiplying throughout by -nx~" " g i v i n g

-nx " x' + n(a + U " ' U ' ) x " = bn,

and then making the transformation y = x "", from which y' = - n x " " " ^ x ' , and the equation becomes

y' + n(a + U " ' U ' ) y = bn Eq. 2,16 The Integrating factor of this equation is

q = exp.n f(a + U " ' U ' ) d t = exp. ant. exp. n logg U(t) = e^"*. U"(t)

and the general solution is

y = U - " ( t ) . e - ^ * [ C + b n j e^''*. u"(t)dt ]

X = y-Wn = u ( t ) I C e - ^ " t + bne"^'^t/ e ^^^^ U"(t)dt 1 Eq. 2.17 The solutions In the various special cases a r e :

Case 1.

5

-+ bn exp [ - n ( a -+ p ) t j / e x p [^(a -+ p)t~| P"(t) dt J Case 2.

x = ( A e ^ ' * + Be^» t . f _, -ant , , -ant f a n t , . \ t _ \ t . n , ^ l ) Ce + bne / e (Ae • + Be * ) dt E q . "dt E q . 2.18

-Vn

2.19 Case 3.

Put p = O in Eq. 2.18. This is the result given in Ref. 1. 3. An application

An interesting application of the previous results lies in their use as examples for testing stability criteria derived by the use of Lyapunov's direct miethod. Consider, for example, the equation

x" + F(x).x' + G(x) = 0 Eq. 3.1 Putting X G(x)dx, o *(x) = ƒ F(x)dx. r (x) = ƒ o o

then Eq . 3 . 1 may be written as the system

x ' = x - * ( x ) ' ) ^ „ „ ' ^ 1 Eq. 3.2

X' = - G ( x )

a 1

The class of differential systems to be discussed are those in which F(x), G(x), * (x) and r(x) are analytic in x and it follows from Ref. 5, p. 34, Theorem 8.1 that under these conditions Eq. 3. 2 is a Lipschitzian system, i . e . , one and only one solution curve passes through each ordinary point of the X , X plane. See also the discussion of integral curves in Ref. 3.

F o r the purpose of establishing the stability of the origin x = 0, x = 0, the scalar function

V(x , x ) = r ( x ) + ix* Eq. 3.3

1 « 1 2

has been used extensively. See Ref. 6, p. 60. The total differential coefficient of V with respect to t is then

V'(x, ,x^) = G(x^). x; + x ^ ^ '

= G(x ) fx^ - « (x )'1 - x^ . G(x^)

6

-(a) • (o) = G(o) = O,

(b) • ( x ) / x > 0. G(x)/x > 0, x M . . (c) r (x) •• « a s II X II •• €«

With t h e s e conditions it i s c l e a r t h a t V(x^, x^) i s p o s i t i v e definite and V^ ( x , , Xg) i s negative definite. It follows t h a t if the conditions (a), (b) and (c) a r e s a t i s f i e d then the o r i g i n i s a s y m p t o t i c a l l y s t a b l e for a r b i t r a r y i n i t i a l x^ , x^ .

Applying t h e s e r e s u l t s to E q . 2 . 1 4 in the p a r t i c u l a r c a s e n = 1, a = 0 g i v e s F(x) = 3bx + k, G(x) = b » x ' + bkx» + e x , • (x) = 3 bx* + kx 2 and r ( x ) = 1^ b ' x * + l ^ b k x ' + l _ c x ' ; 4 3 2 which i m p l i e s that « (o) = G(o) = 0. r ( x ) - m a s II x II - » , and * ( x ) / x = 3^bx + k , 2 G(x)/x = b*x* + bkx + c

Thus the conditions (a) and (c) a r e s a t i s f i e d and condition (b) will be s a t i s f i e d

provided

3 bx + k > 0 (i) 2

and

b ' x ' + bkx + c > 0 (ii) With k sufficiently s m a l l condition (il) can b e s a t i s f i e d for all x ^ 0; h o w e v e r , t h i s

i s not s o for condition ( i ) . T h u s on the b a s i s of t h e Lyapunov function E q . 3 . 3 t h e p r e s e n t equation i s not a s y m p t o t i c a l l y s t a b l e at the o r i g i n for a r b i t r a r y Initial x^, X J. T h i s d o e s n o t , of c o u r s e , m e a n that the a s y m p t o t i c i n s t a b i l i t y of the o r i g i n h a s b e e n e s t a b l i s h e d .

7

-solutions given by Eq. 2.18 and Eq. 2.19. These a r e : Case 1. . P t f X = P ( t ) . e ' = P(t) e" t C + b

ƒ e''*P(t)dtJ

C + b c ' ' e ' ' * ["(Ep +Ptr)Sinfft+ ( F P - E t r ) C o s f f t ] J " ' Eq. 3,5 Case 2. x = (Ae ^ + Be «^) [ C + b j (Ae ' + Be « ) dt J

= (Ae^^* + Be''»*) 1 C + b (AX,'* e <* + BX^-' e * ) j Eq. 3,6 In Case 1 it follows that provided P = -^k < 0, then x-* 0 as t •• <» 1. e. the system is asymptotically stable for arbitrary initial x. In Case 2 X, = -^k - (c - jk») 2 and \ = -jk + (c - ik") 2 and provided k > 0 then the t e r m

Ae'''*[ C+b(AX,-'e'''*+BX,-*e''«SJ'^ 0

as t •• M. The other t e r m - 1 . t 1 ; a

Be * I C + etc. J

also tends to zero as t -. » provided X^ < 0. When X^ > 0 then

X -Be^»*.(bBV' e''«V' = XJb = b'' [-ik + (c - ik') ^] Eq. 3.7

as t •• » . It will be observed that the singular points on the x axis a r e given by G(x) = 0, i . e .

x(b*x* + bkx + c) = 0 which has the three solutions

x = 0, b " ' r-ik± (c -ik*)^"[

The interpretation of Eq. 3. 7 is then that the solution curve has moved into one of the other singular points away from the origin. In Case 1, of course, the only aingular point given by G(x) = 0 is at the origin.

8

-4. Equations of higher order

It is clearly an asset if the method of synthesis can be used to establish equations of order greater than two for which exact solutions a r e known. That this is possible may be seen from the following derivation of an equation of third order.

The starting point is again Eq. 2.1 from which follows Eq. 2.2 and the result

U'^U" = x"' {^x" + x ' . f(x) + X A ' ( X ) J , Eq. 4.1

where f(x) is given by Eq. 2 . 5 . Differentiating Eq. 4.1 with respect to t gives U'^U'" - U ' ' u ' . U ' ' u " = x - ' x " ' - x - ' x ' x " + x ' ' x " f ( x ) + x " ' ( x ' ) ' d f / d x

- x"'(x')*f(x) + 2x'A(x).dA/dx Now

U ' ^ U ' X U " ' u " = x " ' fx'^x' + A(x)~j T x " + x'. f(x) + xA*(x) 1 , and it follows that

U " U " ' = x ' ' [ x ' " + x" |^f(x) + A(x)~] + X' rA»(x) + f(x)A(x) + 2A(x).dA/dx + x'.df/dx~j + xA'(x) J

F r o m these there may be formed the equation

U ' ' [ U'" +^g(t).U" + ^, (t)U' + * ^ u j = x ' ' ( x'" + r^x) + A(x) + ^^(t) 1 x'

+ [A*(x) + f(x) A(x) + 2A(x). dA/dx + ^ (t). f(x) + ^ (t) + x», d f / d x I x' + [A'(x) + ^ J t ) . A " ( x ) + ^^(t).A(x) + ^ J t ) ^ x 1

It follows, in a similar way to the argument in Section 2, that if U(t) is first chosen to satisfy

U'" + 0 (t)U" + ^ (t)U' + ^ (t). U = 0, Eq 4 2

a 1 o

then any solution x(t) of Eq. 2.1 will satisfy

x'" + F ( x . t ) . x " + G(x, X', t ) . x ' +H(x,t) = 0. Eq. 4.3 where

F(x,t) = f(x) + A(x) + V(t)

a

G(x,x»,t) = A"(x) + f(x).A(x) + 2 A ( x ) . d A / d x + A ( t ) . f ( x ) + ^ (t) + x ' . d f / d x 2 1 E q . 4 . 5 and H(x,t) r A ' ( x ) + 4>W.A (x)+ 0 ( t ) . A ( x ) + ^(t)~| X Eq. 4.6 Taking A(x) = bx" - a, then f(x) = (2 + n ) b x " - 2a, n - 1 flA/dx = nbx n - i df/dx = n(2 + n ) b x ; if, in addition, ^ (t) - a , > (t) - /S , ^ . ( t ) - y , 2 1 o

where a, /9 and y are real constants, then the coefficients of Eq. 4.3 become

F(x) = (3 + n)bx" - 3a + a

G(x,x') = (bx" - a)* + (bx" - a) [(2 + n)bx" - 2a J + 2nbx'' ' ' (h^ - a)

+ ar(2 + n)bx" - 2a~] + /9 + n(2 + n ) b x " ' ' .x'

H(x) = ("(bx" - a ) ' + a (bx" - a ) " + fi{hx - a) + y 1 x In the s p e c i a l c a s e a • o t h e s e r e d u c e to F(x) = (3 + n) bx" + a G ( x , x ' ) = (3 + n ) b ' x ' " + 2 n b ' x * " ' ' + (2 + n ) b o x " + /? + n(2 + n ) b x " " V x ' H(x) = b ' x " ' + V « b V ' ' ^ ' ^ fihx"^' + y x

10

-References

1. Smith, R.A. xA. simple non-linear oscillation. Journal of the London Math. Soc. , Vol. 36, pp. 33-34, 1961.

2. Stoker, J . J . Nonlinear vibrations in mechanical and electrical systems.

Interscience Publishers, New York, 1950, 3. Christopher, P . A . T . The stability of the short-period motion of

an airframe having non-linear normal force and pitching moment curves.

The Aeronautical Quarterly, Vol. XI, pp. 255-268, 1960.

4. Kamke, E. Differentialgleichungen Ibsungsmethoden und losungen.

Vol. I, 5th edition, Akademische Verlagsgesell-schaft Beker and E r l e r Kom. -Ges. , Leipzig, 1956.

5. Coddington, E . A . and Theory of ordinary differential equations. Levinson, N. McGraw-Hill (1955).