Ordinary Differential Equations and Calculus of Variations : Book of Problems 9789810221911, 9810221916

Table of contents :
Contents
Preface
Chapter 1. First Order Differential Equations
1.1 Separable equations
1.2 Homogeneous equations
1.2.1 Quasihomogeneous Equations
1.3 Exact equations
1.3.1 Integrating Factors
1.4 Linear equations
1.4.1 Bernoullis Equation
1.4.2 Darboux's Equation
1.4.3 Riccati's Equation
1.4.4 Bool's Equation
1.5 Nonlinear equations
1.5.1 Solvable Equations. General Solution
1.5.2 Solvable Equations. Singular Solution
1.5.3 Unsolvable Equations
1.6 Applications in physics
1.6.1 Mechanics
1.6.2 Hydrodynamics
1.6.3 Electrical Networks
1.6.4 Kinetic Theory
1.6.5 Nuclear Physics
1.6.6 Optics
1.7 Miscellaneous problems
Chapter 2. N-th Order Differential Equations
2.1 Reduction of order
2.1.1 Simple Cases
2.1.2 Homogeneous Equations
2.1.3 Exact Equations
2.1.4 Linear Equations
2.1.5 The Initial Value Problem
2.2 Linear homogeneous equations
2.2.1 Exponential Solution
2.2.2 Power Solution
2.2.3 Transformations of Equation
2.2.4 The Initial Value Problem
2.3 Linear nonhomogeneous equations
2.3.1 Method of Variation of Parameters
2.3.2 Method of Undetermined Coefficients
2.3.3 The Influence Function
2.3.4 The Initial Value Problem
2.4 Linear equation with constant coefficients
2.4.1 The Homogeneous Equation with Constant Coefficients
2.4.2 The Complete Equation with Constant Coefficients. Method of Undetermined Coefficients
2.4.3 The Method of Variation of Parameters
2.4.4 Symbolic Methods
2.4.5 Laplace Transform
2.5 Equations with polynomial coefficients
2.5.1 Changes of Variable
2.5.2 Substitutions
2.5.3 Substitutions and Changes of Variable
Chapter 3. Linear Second Order Equations
3.1 Series solutions
3.1.1 Ordinary Point
3.1.2 Regular Singular Point
3.1.3 Irregular Singular Point
3.2 Linear boundary value problem
3.2.1 Homogeneous Problem
3.2.2 Nonhomogeneous Problem
3.2.3 Green's Function
3.3 Eigenvalues and eigenfunctions
3.3.1 Self-adjoint Problems
3.3.2 The Sturm-Liouville Problem
3.3.3 Nonhomogeneous Problem
Chapter 4. Systems Of Differential Equations
4.1 Linear systems with constant coefficients
4.1.1 Homogeneous Systems
4.1.2 Homogeneous Systems. Euler's Method
4.1.3 Euler's Method. Different Eigenvalues
4.1.4 Euler's Method. Repeated Eigenvalues
4.1.5 Repeated Eigenvalues. Method of Associated Vectors
4.1.6 Repeated Eigenvalues. Method of Undetermined Coefficients
4.1.7 Homogeneous Systems. Matrix Method
4.1.8 Nonhomogeneous Systems
4.1.9 Method of Variation of Parameters
4.1.10 Method of Undetermined Coefficients
4.1.11 Matrix Method
4.1.12 Initial Value Problem
4.1.13 Laplace Transform
4.1.14 Systems of Higher Order Equations
4.2 Linear systems
4.2.1 Solution by Eliminations
4.2.2 Matrix Method
4.2.3 Nonhomogeneous Linear Systems
4.2.4 Initial Value Problem
4.3 Nonlinear systems
4.3.1 Method of Eliminations
4.3.2 Method of Integrable Combinations
4.3.3 Systems of Bernoulli's Form
4.3.4 Method of Complex Variable
4.3.5 Systems of Canonical Form
Chapter 5. Partial Equations of the First Order
5.1 Linear partial equations
5.2 Pfaffian equation
5.2.1 Mayer's Method
5.3 Nonlinear partial equations
5.3.1 Lagrange Charpit's Method
Chapter 6. Nonlinear Equations And Stability
6.1 Phase plane. Linear systems
6.2 Almost linear systems
6.3 Liapunov's second method
Chapter 7. Calculus of Variations
7.1 Euler's equation
7.2 Conditional extremum
7.2.1 Isoperimetric Problem
7.3 Movable end points
7.4 Bolza problem
7.5 Euler-Poisson equation
7.6 Ostrogradsky equation
Chapter 8. Answers To Problems
A1.1 Separable equations
A1.2 Homogeneous equations
A1.3 Exact equations
A1.4 Linear equations
A1.5 Nonlinear equations
A1.6 Applications in physics
A1.7 Miscellaneous problems
A2.1 Reduction of order
A2.2 Linear homogeneous equations
A2.3 Linear nonhomogeneous equations
A2.4 Linear equation with constant coefficients
A2.5 Equations with polynomial coefficients
A3.1 Series solutions
A3.2 Linear boundary value problems
A3.3 Eigenvalues and eigenfunctions
A4.1 Systems with constant coefficients
A4.2 Linear systems
A4.3 Nonlinear systems
A5.1 Linear partial equations
A5.2 Pfaffian equation
A5.3 Nonlinear partial equations
A6.1 Phase plane. Linear systems
A6.2 Almost linear systems
A6.3 Liapunov's second method
A7.1 Euler's equation
A7.2 Conditional extremum
A7.2.1 Isoperimetric problem
A7.3 Movable end points
A7.4 Bolza problem
A7.5 Euler-Poisson equation
A7.6 Ostrogradsky equation
Bibliography
1-15
16-33
34-51
Index

Citation preview

Ordinary Differential Equations and Calculus of Variations

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ORDINARY DIFFERENTIAL EQUATIONS AND CALCULUS OF VARIATIONS Book of Problems

M. V. Makarets Kiev T. Shevchenko University, Ukraine

V. Yu. Reshetnyak Institute of Surface Chemistry, Ukraine

World Scientific V h

Singapore

• New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pie. Lid. PO Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, Rivet Edge, NJ 07661 UK office: 57 Shelton Street, Coven! Garden. London WC2H 9HE

ORDINARY DIFFERENTIAL EQUATIONS AND CALCULUS OF VARIATIONS Copyright© 1995 by World Scientific Publishing Co. Pte. Ud. All rights reserved. This book, or parrs thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of materia] in (his volume, please pay a copying fee through the Copyright Clearance Center, Inc.. 37Congress Street. Salem. MA 01970. USA.

ISBN 981-G2-2191-6

This book is printed on acid-free paper

Printed in Singapore by Uto-Prinl

Contents PREFACE

ix

1

FIRST ORDER D I F F E R E N T I A L EQUATIONS 1 1.1 Separable equations . . . . 1 1.2 Homogeneous equations 9 1.2.1 Quasi homogeneous Equations . . . . 16 1.3 Exact equations . 19 1.3.1 Integrating Factors . 2 5 1.1 Linear equations . . . 33 1.4.1 Bernoulli's Equation . 4 1 1.4.2 Darboux's Equation . . 44 1.4.3 Riccati's Equation . 46 1.4.4 Bool's Equation . . . . 50 1.5 Nonlinear equations . . . . ... 52 1.5.1 Solvable Equations. General Solution . . 53 1.5.2 Solvable Equations. Singular Solution . . . 59 1.5.3 Unsolvable Equations . . . . . . . 6 2 1.6 Applications in physics . . . . 64 1.6.1 Mechanics ... . . . . . . . 64 1.6.2 Hydrodynamics . 67 1.6.3 Electrical Networks . 68 1.6.4 Kinetic Theory . 69 1.6.5 Nuclear Physics . 7 2 1.6.6 Optics . . . . 7 2 1.7 Miscellaneous problems . . . . 74

2

N-th ORDER DIFFERENTIAL EQUATIONS 2.1 Reduction of order . . . 2.1.1 Simple Cases . . 2.1.2 Homogeneous Equations 2.1.3 Exact Equations . 2.1.4 Linear Equations 2.1.5 The Initial Value Problem 2.2 Linear homogeneous equations ... ...

. . .

.

77 77 78 79 80 82 83 87

vi

CONTENTS

2.3

2.4

2.5

3

4

2.2.1 2.2.2 2.2.3 2.2.4 Linear 2.3.1 2.3.2 2.3.3 2.3.4 Linear 2.4.1 2.4.2

Exponential Solution . . . 89 Power Solution . . . . 90 Transformations of Equation . . 92 The Initial Value Problem 94 nonhomogeneous equations . 97 Method of Variation of Parameters . . 98 Method of Undetermined Coefficients 100 The Influence Function . 102 The Initial Value Problem . 103 equation with constant coefficients . 107 The Homogeneous Equation with Constant Coefficients. 107 The Complete Equation with Constant Coefficients. Method of Undetermined Coefficients. . . . 112 2.4.3 The Method of Variatiou of Parameters . . .. 120 2.4.4 Symbolic Methods . . 123 2.4.5 Laplace Transform . . . 131 Equations with polynomial coefficients 140 2.5.1 Changes of Variable . 141 2.5.2 Substitutions . .143 2.5.3 Substitutions and Changes of Variable 145 2.5.4 Series Solutions 146

L I N E A R SECOND ORDER EQUATIONS 3.1 Series solutions 3.1.1 Ordinary Point . . 3.1.2 Regular Singular Point 3.1.3 Irregular Singular Point 3.2 Linear boundary value problem 3.2.1 Homogeneous Problem 3.2.2 Nonhomogeneous Problem 3.2.3 Green's Function . 3.3 Eigenvalues and eigenf unctions 3.3.1 Self-adjoint Problems 3.3.2 The Sturm-Llouville Problem 3.3.3 Nonhomogeneous Problem

.

.

SYSTEMS OF D I F F E R E N T I A L E Q U A T I O N S 4.1 Linear systems with constant coefficients 4.1.1 Homogeneous Systems . 4.1.2 Homogeneous Systems. Euler's Method 4.1.3 Euler's Method. Different Eigenvalues 4.1.4 Euler's Method. Repeated Eigenvalues 4.1.5 Repeated Eigenvalues. Method of Associated Vectors

153 153 153 157 166 172 173 175 178 182 184 186 188 191 191 191 192 192 193 194

CONTENTS

5

4.1.6 Repeated Eigenvalues. Method of Undetermined Coefficients . 4.1.7 Homogeneous Systems. Matrix Method . • • • • 4.1.8 Nonhomogeneous Systems 4.1.9 Method of Variation of Parameters . . . . 4.1.10 Method of Undetermined Coefficients . 4.1.11 Matrix Method . . . . 4.1.12 Initial Value Problem . . . 4.1.13 Laplace Transform . . . 4.1.14 Systems of Higher Order Equations 4.2 Linear systems . 4.2.1 Solution by Eliminations . 4.2.2 Matrix Method 4.2.3 Nonhomogeneous Linear Systems . . . . . 4.2.4 Initial Value Problem 4.3 Nonlinear systems . . . . . . 4.3.1 Method of Eliminations 4.3.2 Method of Integrable Combinations. 4.3.3 Systems of Bernoulli's Form 4.3.4 Method of Complex Variable . . . . . 4.3.5 Systems of Canonical Form . . . . . .

198 199 203 203 204 205 206 207 208 216 216 219 219 221 224 225 228 230 231 232

PARTIAL EQUATIONS OF T H E FIRST ORDER

237

5.1 5.2 5.3

6

Linear partial equations Pfaffian equation . 5.2.1 Mayer's Method. . . . Nonlinear partial equations 5.3.1 Lagrange - Charpit's Method

NONLINEAR

6.1 6.2 6.3 7

vii

. . . . . . .

. . . . .

. . . .

EQUATIONS AND STABILITY

Phase plane. Linear systems Almost linear systems Liapunov's second method

255

. . . .

.

. . . .

. . . .

.

CALCULUS O F VARIATIONS

7.1 7.2 7.3 7.4 7.5 7.6

237 244 .246 248 250

. . . .

Euler's equation Conditional extremum 7.2.1 Isoperimetric Problem . . . Movable end points Bolza problem Euler-Poisson equation . . . . Ostrogradsky equation .

257 266 273 279

. .

.

.

. .

. .

279 284 . 2 8 8 292 . . 299 . . . 301 . . 303

viii 8

CONTENTS ANSWERS TO PROBLEMS 8.1 Separable equations . 8.2 Homogeneous equations • 8.3 Exact equations . . . • 8.4 Linear equations 8.5 Nonlinear equations 8.6 Applications in physics . . . 8.7 Miscellaneous problems . . . 8.8 Reduction o( order 8.9 Linear homogeneous equations . 8.10 Linear nonhomogeneous equations . ... 8.11 Linear equation with constant coefficients. . 8.12 Equations with polynomial coefficients 8.13 Series solutions . 8.14 Linear boundary value problems 8.15 Eigenvalues and eigenfunctions 8.16 Systems with constant coefficients . 8.17 Linear systems . . . . . . . 8.18 Nonlinear systems . ... 8.19 Linear partial equations . 8.20 Pfaffian equation . 8.21 Nonlinear partial equations. . . 8.22 Phase plane. Linear systems . . 8.23 Almost linear systems . 8.24 Liapunov's second method . . ... . 8.25 Euler's equation . . . . . . . 8.26 Conditional extremum . 8.27 Isoperimetric problem . . 8.28 Movable end points . . 8.29 Bolza problem . . ... 8.30 Euler-Poissou equation . . . 8.31 Ostrogradsky equation. . . . .

307 307 308 310 312 315 318 321 323 • • 326 327 330 336 338 342 344 346 350 351 . 354 355 355 . 357 358 359 . 359 . 360 361 361 362 362 363

BIBLIOGRAPHY

365

INDEX

369

Preface This problem book contains exercises for courses in differential equations and calculus of variations at universities and technical institutes. It is designed for no n-mat hematics students and also for scientists and practicing engineers who feel a need to refresh their knowledge of such an important area of higher mathematics as differential equations and calculus of variations. Each section of the text begins with a summary of basic facts. This is followed by detailed solutions of examples and problems. The book contains more than 260 examples and about 1400 problems to be solved by the students, a considerable part of which have been composed by the authors themselves. Numerous references are given at the end of the book. These furnish sources for detailed theoretical approaches, and expanded treatment of applications. In preparing this book for publication, Mr. Y.-S. Kim rendered a great help to us.

ix

Chapter 1

FIRST ORDER DIFFERENTIAL

1.1

EQUATIONS

Separable equations

A differential equation which can be written in the form M(x)dx + N(y)dy = 0,

(1)

where M is a function of X alone and N is a function of y alone, is said to be separable. The solution is j M{x)dx + j N(y)dy = C, (2) where C is an arbitrary constant. The problem is then reduced to the problem of evaluating the two integrals in (2). In Eq.(l) we say that the variables are separated. Example 1. Find the solution of the equation y' = e'*> which is such that y — 0 when x = 0. The equation may be written as y' =

eV,

from which i t is seen that the separated form is e~*dy — e'dx. Integrating now gives the general solution z

-e~> = t + C, and we have to find the value of the constant C such that x and y vanish simultaneously. On putting t = y — 0, we have — 1 = 1 + C whence C = —2. The appropriate solution is given by e"* = 2 - e' 1

CHAPTER

2

1. FIRST

ORDER DIFFERENTIAL

EQUATIONS

Example 2. Solve the equation xydx+(x+l)dy

= 0.

(3)

If y j£ 0 and x + 1 ^ 0, we can divide by tj and i + I and put the equation in the form dy

xdx

= 0.

Integrating, J

y

J x + 1

M j r l + i - l n \x + l\ = C. Taking exponential of both sides yields 1

jz-d^

+ l j e " , C, = In |C|.

Equation (3) has also solutions y = 0 and x = — 1 The first one can be obtained from the general solution when arbitrary constant C\ — 0 and therefore JJ = 0 is the particular solution. The second solution x — — 1 can't be obtained from the genera! solution and therefore x = — 1 is the singular solution. Then the solution of the problem (3) is y = C,(x+ \)e-*

if x jt - 1 ; also I = - 1 .

Example 3. Solve the initial value problem ,

y c o t i + y = 2; y ( j ] = 0. KJf

(4)

2 and cot x ^ 0 the differential equation can be written as dy

• + tan xdx — 0.

Integrating, f dy I y—2

l sin xdx _ ^ J cos x '

l n | s - 2 | - l n | e o s i | = C. Whence B = 2+&eosjf

)

(5)

where C, = ]a\G\ is an arbitrary constant. To determine the particular solution satisfying the prescribed initial condition we substitute x = TT/3 and y = 0 into

1.1. SEPARABLE

EQUATIONS

3

Eq,(5)> obtaining Ci — —4. Hence the desired particular solution is given explicitly by y- 2-

icosx.

Example 4. Solve the equation 2

3

3

2

y ( r + l ) dx + (x - 5x + 6x) dy = 0. If y j i 0 and x ^ 0, 2, 3 then the separated form is 3

dy If

(x + l)dx x -Sx + 6x

2

3

or

2

2

Sx - 6x + 1 x - ^ + 6x } " 3

Using partial fractions we can write a

i

a

5 i - 6 i + 1 _ 5 i - 6x + 1 _ A - 5 i + 6 i ~ s.(g - 2)(x - 3} ~ x 3

2

+

B x-2

'

C 1-3'

Multiplying this by i we find Cx 5 I - fa + 1 ( i - 2)(z - 3) = 4 + z - 2 +••- g - 3 2

Then letting i —< 0, we have 2

5i - 6i + 1 [x - 2)(x - 3) Similarly, multiplying by I - 2 and letting x — 2 yields 2

B

=

5x - 6x + 1 * ( * - 3)

and multiplying by x - 3 and letting i — 3 yields C = 28/3, Putting these values, we get 28 1 j/ I 6 i 2 i - 2 3 i - 3 dx which, on integration, gives 5

i i = y

r

+

l l 28 + i l n i i | - ^ l Q | x - 2 | + - l n | i - 3 | + C, 6 2 J

where C is an arbitrary constant. The given equation has also the singular solutions y = 0, (a; # 0, x * 2, a; # 3), s = 0 (y ^ 0), x = 2 (J j i 0) and x = 3 (y # 0).

CHAPTER

4

I . FIRST

ORDER DIFFERENTIAL

EQUATIONS

Example 5. Solve the equation 3

(x + \)dy - ydx = 0. If i £ - 1 and g / O w t can write this equation in the form dy

3

dx 3

7

~ * +l '

J

Since i + 1 = ( i + l ) ( i — i + 1), we have using partial fractions 1 + i

A

x+1

+

Bx + C x - x + r a

2

where A, S, C are undetermined constants. Multiplying by (x + 1 )(x - i + 1) we obtain 1 = A {x* - x + l ) + {Bx + C)(x + 1) = x'(-4 +

B) +

x{-A + B + C) + [A + C).

Since this is an identity we have on equating coefficients of like powers of x, A +B = 0 -A-i-B

t

+ C = 0,

A + C = I. Solving these we find A = 1/3, B - - 1 / 3 , C - 2/3. On substituting these values and integrating we have

/ = I | 3

n

| '

I

+

f dy _ 1 / _rfx_ _ 1 / S 3/x + l 3/

x-2 + l

l l _ I f ~ ' / 2 ) d ( x - 1/2) 1 r dx 3/ ( x - l / 2 ) + 3/4 2 J ( x ~ 1 / 2 ) ' + 3/4 1

J

+

!

= ^ In |x + 11 - i In ( x - x + 1) + -L arctan

+ fj,

or 5

, I | 1 . (x + 1 ) 1 2x - 1 _ in ji = - In — — + - = arctan — 7 = - + C, 6 x - x +1 ,/3 1

where C is an arbitrary constant. The original equation has also the singular solutions x = - l ( ! 0 ) and = 0 ( x ^ - l ) . If a differential equation can be written in the form y ?

V

y' = f[ax + 63,)

(6]

1.1. SEPARABLE

EQUATIONS

then we put z — ax + by and have

or

in which the variables are separable. Example 6. Solve the equation [x + y + l)dx + {2x + 2y-\)dy

= 0.

If 2(x + j/) — 1 ^ 0 the differential equation may be written as ,

(« + y) + i 2{x + y) — \

=

y

Put x + y — z , so that rifx =

1+

f( i i

Thus z is determined as a function of x by an equation of the form (*z_ _z + l dx 2z - 1 1

z-2 2z-V

which is separable 2x- 1 z-2 or

Integrating,

= dz, zjt 2, 3 z-2

02 — dx.

/K-H>*-/* 2z + 3 1 n | z - 2 | = x + C

from which, on putting z = x + j/ , we have the general solution 2(x + j,) + 31n|x + y - 2| = x + C, or x + 2j, + 31n |x + 3 , - 2 | = C, where C is an arbitrary constant. The original equation has also the singular solution z — 2 or y = 2

CHAPTER

I . FIRST

ORDER DIFFERENTIAL

EQUATIONS

Example 7. Solve d By putting z = ix + 2y, we have z' = 4 + 2y' = 4 + 2 V 7 ^ T whence

, 4 + 2VJ^T

Integrating, / + («V -»)»*••. M^y)(y + z ) dy + (x - xy]dx = 0, n = p(x 4- y ) . ( r + y ) d x - x d y « 0. ( x y + y ) d x - x d y = 0. (2x y*-y)dx + {2x y -x)dy = 0. xy dx + ( x y - x)dy = 0. ( z + y + 1)dx - 2xydy = 0,^ = fi{x ± y ). 2

2

3

3

3

3

1

2

1

3

2

1

2

2

Linear equations

The general form of a first order iinear differential equation is A(x)^

+ Blx)y + C(x) = 0.

On division by the first coefficient, it can be put in the form

g + P(x)y = Q(x,.

(1)

If the second member is zero, Q(x) - 0, the equation can be solved by separating the variables ^ ^

+ />(x)y = 0, l n | y | = - j P(x)dx + C,

= -Pdx, y=

C exp[-J

where Ci is an arbitrary constant.

(2)

t

P(x)dx\,

(3)

34

CHAPTER I . FIRST

ORDER DIFFERENTIAL

EQUATIONS

ID general case (1} we multiply Eq.(l) by exp|/ P(x) ( x ) d x ] + e x p [ - / P(x)dx] j e x p [ | P(x)dx] where C is an arbitrary constant. Example 1. Solve the differential equation d

1

y , dx x

i

Here P ( i } = 1/x, and we multiply by exp [/ P(x)dx\ = x, ^

, r

whence, integrating,

or 3

y = -x + - . 4 x Example 2. Find the solution of the initial value problem j/-2xj, = i ;

y(0) = l .

Here P{x) = -2x, j R(x)dx = - j2xdx M

Q(x)d , X

1.4. LINEAR

EQUATIONS

35

Hence multiplying by e **, we have e.-*(y'-2xy)

= xe-*'

so that 1

(ye- )'

=

«."-

Therefore 1

y e " ^ = /are"" 'rf*+ & = - ^ e ^ + G and finally

To satisfy the initial condition y(0) - I w e must choose C — 3/2. Hence 1 y

=

-2

, +

3

2

J* e

is a solution of the given initial value problem. T H E O R E M i . Ify = U{x) is a particular solution oj (I) and ify = V(x) is a particular solution of (2), then y = CV(x) + U[x) is the genera! solution of (I). We see from this Theorem that if we notice or can find in any way a particular solution of (1), the problem is then reduced to the solution of the less complicated Eq.(2), the general solution of (1) can be written down at once. Example 3. Solve the differential equation y' + y tanx — tanx. We note that y = 1 is a solution. The equation y' + y tan x = 0 can be solved by separating the variables dy y

sin x ^ cos x

or In \y\ = In | cos x|. Taking exponential of both sides yields y — cos x. The general solution then is y = C cos x + 1.

36

CHAPTER

I. FIRST

ORDER

DIFFERENTIAL

EQUATIONS

T H E O R E M 2. / / y = U{x) and y = V(x) are different particular solutions of (1), then Ike general solution is 9 = G[V(*)-V(x)]

+ V(x).

This can be put in various forms. Since we have also y — Oj [U[x]

V ( i ) j + U(x).

Example 4. Solve the differential equation 1 x - 1T f

V

1 =

—

x

1

This equation is found to be satisfied by y — 1 and by y = x. The general solution is then 5, = (7(i - 1) + 1. Consider the following method of solving the general linear differential equation of the first order (1). If Q{z) is not identically zero, then we assume that the solution is of the form (3) y = A(x)exp[-

J P(x)dx\

(6)

where an arbitrary constant C\ is now a function of x and exp [— f P(x)dx\ is a solution of corresponding homogeneous equation. By substituting for t; in the given differential equation we have A'{x)exp

[- j P( )dx\-A(x)P(x)exp

[- /

X

+P(xM(*)exp [- j P(x)dxj

P{x)dx\

=Q[x),

or =