TOPICS IN

DIFFERENTIAL EQUATIONS AND INTEGRAL TRANSFORMS

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TOPICS IN

DIFFERENTIAL EQUATIONS AND INTEGRAL TRANSFORMS

By PARMANAND GUPTA B.Sc. (Hons.), M.Sc. (Delhi) M.Phil (KU), Pre. Ph.D. (IIT Delhi) Ex. Associate Professor and Head, Dept. of Mathematics Indira Gandhi National College, Ladwa & Member, Faculty of Sciences Kurukshetra University Haryana

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CONTENTS Chapter

Pages SECTION I

1. Series Solution of Differential Equations ..................................................................... 3–47 1.1. Introduction ................................................................................................................................. 3 1.2. Power Series ................................................................................................................................ 3 1.3. Convergence of Power Series ..................................................................................................... 3 1.4. Operations on Power Series ....................................................................................................... 4 1.5. Shifting of Summation Index ..................................................................................................... 5 1.6. Analytic Function ....................................................................................................................... 9 1.7. Ordinary and Singular Points ................................................................................................. 10 1.8. Series Solution of Differential Equations ............................................................................... 13 1.9. Power Series Method ................................................................................................................ 13 1.10. Frobenius Method ..................................................................................................................... 27 1.11. Frobenius Method (Continued) ................................................................................................ 36

2. Legendre Equation ....................................................................................................... 48–72 2.1. Introduction ............................................................................................................................... 48 2.2. Legendre Equation and its Solution ....................................................................................... 48 2.3. Legendre Polynomials .............................................................................................................. 51 2.4. Alternative Form of Legendre Polynomial ............................................................................. 54 2.5. Generating Function of a Sequence ........................................................................................ 58 2.6. Generating Function of the Sequence of Legendre Polynomials .......................................... 58 2.7. Recurrence Relations ................................................................................................................ 62 2.8. Rodrigue’s Formula .................................................................................................................. 65 2.9. Orthogonality of Functions ...................................................................................................... 67 2.10. Orthogonality of Legendre Polynomials ................................................................................. 69

3. Bessel Equation .......................................................................................................... 73–101 3.1. Introduction ............................................................................................................................... 73 3.2. Bessel Equation and its Solution ............................................................................................. 73 3.3. Bessel Functions ....................................................................................................................... 77 3.4. General Solution of Bessel Equation in Terms of Bessel Functions .................................... 79 3.5. Particular Forms of Bessel Functions ..................................................................................... 85 3.6. Elementary Bessel Functions .................................................................................................. 85 3.7. Generating Function of the Bessel Functions ........................................................................ 90

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Chapter

Pages

3.8. Recurrence Relations ................................................................................................................ 95 3.9. Orthogonality of Bessel Functions .......................................................................................... 98

4. Hypergeometric Equation ........................................................................................ 102–114 4.1. Introduction ............................................................................................................................. 102 4.2. Hypergeometric Equation and its Solution .......................................................................... 102 4.3. Hypergeometric Functions ..................................................................................................... 104 4.4. Factorial Function .................................................................................................................. 105 4.5. Alternative Form of Hypergeometric Function .................................................................... 105 4.6. Particular Solution of Hypergeometric Equation ................................................................ 106 4.7. Properties of Hypergeometric Functions .............................................................................. 109 4.8. Gauss Theorem ....................................................................................................................... 112 4.9. Vandermonde Theorem .......................................................................................................... 112

5. Sturm-Liouville Problems ........................................................................................ 115–125 5.1. Introduction ............................................................................................................................. 115 5.2. Sturm-Liouville Problem ........................................................................................................ 115 5.3. Eigenvalue and Eigenfunction ............................................................................................... 116 5.4. Reality of Eigenvalues ............................................................................................................ 116 5.5. Orthogonality of Eigenfunctions of a Sturm-Liouville Problem ......................................... 122

SECTION II 6. Partial Differential Equations ................................................................................... 129–140 6.1. Introduction ............................................................................................................................. 129 6.2. Definition of a Partial Differential Equation ....................................................................... 129 6.3. Order of a Partial Differential Equation .............................................................................. 129 6.4. Linear Partial Differential Equation .................................................................................... 129 6.5. Notation ................................................................................................................................... 130 6.6. Formation of a Partial Differential Equation ....................................................................... 130 6.7. Formation of a Partial Differential Equation by Elimination of Arbitrary Constants ..... 130 6.8. Formation of a Partial Differential Equation by Elimination of Arbitrary Functions ..... 135

7. Partial Differential Equations of the First Order (Equations Linear in p and q) 141–156 7.1. Introduction ............................................................................................................................. 141 7.2. Solution of a Partial Differential Equation .......................................................................... 141 7.3. Complete Solution ................................................................................................................... 141 7.4. Particular Solution ................................................................................................................. 142 7.5. Singular Solution .................................................................................................................... 142 7.6. General Solution ..................................................................................................................... 142 7.7. Lagrange Linear Equation ..................................................................................................... 143 7.8. Solution of Lagrange Linear Equation ................................................................................. 143

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Chapter

Pages

8. Partial Differential Equations of the First Order (Equations Non-linear in p and q) ........................................................................... 157–187 8.1. Introduction ............................................................................................................................. 157 8.2. Special Type I : Equations Containing Only p and q ........................................................... 157 8.3. Special Type II : Equations of the Form z = px + qy + g(p, q) ............................................. 162 8.4. Special Type III : Equations Containing Only z, p and q .................................................... 167 8.5. Special Type IV : Equations of the Form f1(x, p) = f2(y, q) ................................................... 172 8.6. Use of Transformations .......................................................................................................... 178 8.7. Charpit’s General Method of Solution .................................................................................. 181

9. Homogeneous Linear Partial Differential Equations with Constant Coefficients ............................................................................................... 188–208 9.1. Introduction ............................................................................................................................. 188 9.2. Partial Differential Equations of Second and Higher Order .............................................. 188 9.3. Homogeneous Linear Partial Differential Equations with Constant Coefficients ............ 188 9.4. Some Theorems ....................................................................................................................... 189 9.5. General Solution of Homogeneous Linear Partial Differential Equation f(D, D′)z = 0 with Constant Coefficients ............................................................................... 190 9.6. General Solution of Homogeneous Linear Partial Differential Equation f(D, D′)z = F(x, y) with Constant Coefficients ....................................................................... 194 9.7. Particular Integral of f(D, D′)z = F(x, y) ................................................................................ 194 9.8. Particular Integral when F(x, y) is Sum or Difference of Terms of the form xmyn ............ 194 9.9. Particular Integral when F(x, y) is of the Form f(ax + by) .................................................. 196 9.10. General Method of Finding Particular Integral ................................................................... 203

10. Non-homogeneous Linear Partial Differential Equations with Constant Coefficients ............................................................................................... 209–225 10.1. Introduction ............................................................................................................................. 209 10.2. Non-homogeneous Linear Partial Differential Equations with Constant Coefficients ... 209 10.3. Reducible and Irreducible Non-homogeneous Linear Partial Differential Equations with Constant Coefficients ..................................................................................................... 209 10.4. General Solution of Reducible Non-homogeneous Linear Partial Differential Equation f(D, D′)z = 0 with Constant Coefficients ............................................................... 210 10.5. General Solution of Irreducible Non-homogeneous Linear Partial Differential Equation f(D, D′)z = 0 with Constant Coefficients ............................................................... 213 10.6. General Solution of Non-homogeneous Linear Partial Differential Equation with Constant Coefficients .............................................................................................................. 216 10.7. Particular Integral of f(D, D′)z = F(x, y) ................................................................................ 216 10.8. Particular Integral when F(x, y) is Sum or Difference of Terms of the Form xmyn ........... 216 10.9. Particular Integral when F(x, y) is of the Form eax+by .......................................................... 219 10.10. Particular Integral when F(x, y) is of the Form sin(ax + by) or cos(ax + by) ..................... 221 10.11. Particular Integral when F(x, y) is of the Form eax+by V(x, y) .............................................. 223

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Chapter

Pages

11. Partial Differential Equations Reducible to Equations with Constant Coefficients ................................................................................................................ 226–231 11.1. Introduction ............................................................................................................................. 226 11.2. Reducible Linear Partial Differential Equations with Variable Coefficients .................... 226 11.3. Solution of Reducible Linear Partial Differential Equations with Variable Coefficients 226

12. Monge’s Methods ...................................................................................................... 232–246 12.1. Introduction ............................................................................................................................. 232 12.2. Partial Differential Equation of Second Order ................................................................... 232 12.3. Intermediate Integral ............................................................................................................. 232 12.4. Monge’s Methods ..................................................................................................................... 232 12.5. Monge’s Method of Solving Rr + S + Tt = V .......................................................................... 233 12.6. Monge’s Method of Solving Rr + Ss + Tt + U(rt – s2) = V .................................................... 241

SECTION III 13. Laplace Transforms .................................................................................................. 249–286 13.1. Introduction ............................................................................................................................. 249 13.2. Laplace Transform of a Function .......................................................................................... 249 13.3. Laplace Transforms of Elementary Functions ..................................................................... 250 13.4. Linearity of the Laplace Transform ...................................................................................... 255 13.5. Shifting Theorems .................................................................................................................. 259 13.6. First Shifting Theorem ........................................................................................................... 259 13.7. Unit Step Function ................................................................................................................. 263 13.8. Second Shifting Theorem ....................................................................................................... 263 13.9. Change of Scale Property ....................................................................................................... 265 13.10. Piecewise Continuous Function ............................................................................................. 269 13.11. Existence Theorem for Laplace Transforms ......................................................................... 269 13.12. Laplace Transforms of Derivatives ....................................................................................... 270 13.13. Laplace Transforms of Integrals ........................................................................................... 275 13.14. Differentiation of Laplace Transforms .................................................................................. 278 13.15. Integration of Laplace Transforms ........................................................................................ 282

14. Inverse Laplace Transforms .................................................................................... 287–329 14.1. Introduction ............................................................................................................................. 287 14.2. Inverse Laplace Transform of a Function ............................................................................. 287 14.3. Existence and Uniqueness of Inverse Laplace Transform .................................................. 287 14.4. Elementary Inverse Laplace Transform Formulae ............................................................. 288 14.5. Linearity of the Inverse Laplace Transform ........................................................................ 290 14.6. Value of L–1(F(s – a)) in Terms of L–1 (F(s)) .......................................................................... 292 14.7. Value of L–1 (e–as F(s)) in Terms of L–1 (F(s)) ........................................................................ 296

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Chapter

Pages

14.8. Value of L–1 F(s/a)) in Terms of L–1 (F(s)) ............................................................................. 297 14.9. Value of L–1 (F(s)/s) in Terms of L–1 (F(s)) ............................................................................ 300 14.10. Value of L–1 (F′(s)) in Terms of L–1(F(s)) ............................................................................... 301 14.11. Value of L−1

F GH

z

∞

s

I JK

F( s ) dS in Terms of L–1 (F(s)) ................................................................... 303

14.12. Convolution Theorem ............................................................................................................. 306 14.13. Inverse Laplace Transforms by the Method of Partial Fractions ............................... 313 14.14. Solution of Differential Equations by Using Laplace Transformation .............................. 324

15. Solution of Integral Equations Using Laplace Transformation ........................... 330–336 15.1. Introduction ............................................................................................................................. 330 15.2. Definition of Integral Equation ............................................................................................. 330 15.3. Method of Solving Integral Equation of Convolution Type ................................................. 330 15.4. Integro-differential Equation ................................................................................................. 333

16. Solution of Systems of Differential Equations Using the Laplace Transformation .......................................................................................................... 337–346 16.1. Introduction ............................................................................................................................. 337 16.2. Method of Solving System of Differential Equations ........................................................... 337

17. Fourier Transforms ................................................................................................... 347–385 17.1. Introduction ............................................................................................................................. 347 17.2. Fourier’s Integral Theorem .................................................................................................... 347 17.3. Fourier Transform and Its Inverse ....................................................................................... 348 17.4. Shifting Property of Fourier Transforms .............................................................................. 349 17.5. Modulation Property of Fourier Transforms ........................................................................ 350 17.6. Convolution Theorem ............................................................................................................. 350 17.7. Fourier Sine and Cosine Transforms .................................................................................... 358 17.8. Linearity of Transforms ......................................................................................................... 377 17.9. Change of Scale Property of Transforms .............................................................................. 378 17.10. Transforms of Derivatives ...................................................................................................... 379 17.11. Parseval’s Identities ............................................................................................................... 380

18. Solution of Differential Equations Using Fourier Transforms ............................. 386–397 18.1. Introduction ............................................................................................................................. 386 18.2. Partial Differential Equation ................................................................................................. 386 18.3. Method of Solving Partial Differential Equation by Using Fourier Transforms .............. 387

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SECTION I

SERIES SOLUTION OF DIFFERENTIAL EQUATIONS

1

Series Solution of Differential Equations 1.1. INTRODUCTION

We know that linear differential equations with constant coefficients can be solved and their solutions are elementary functions.* There also exists some methods in which linear differential equations (with some specific types of variable coefficients) can be solved and their solutions are elementary functions. In general, a linear differential equation with variable coefficients may not admit of any solution which is expressible in terms of elementary functions. In the present chapter, we shall learn the method of solving linear differential equations and obtaining solutions in the form of an infinite series. 1.2. POWER SERIES ∞

An infinite series of the form

∑a

m (x

− x0 ) m = a0 + a1 ( x − x0 ) + a2 ( x − x0 ) 2 + ...... is called a

m=0

power series in x – x0. Here a0, a1, a2, ...... are constants, called the coefficients of the power series. The constant x0 is called the centre of the power series. In the above power series, x is a variable. ∞

In particular, if x0 = 0, then the power series in x is 2

∑a

mx

m

= a 0 + a1 x + a 2 x 2 + ...... .

m=0

3

( x − 2) ( x − 2) + + ...... is a power series in x – 2. 2! 3! The centre of this power series is 2.

For example, 1 + ( x − 2) +

1.3. CONVERGENCE OF POWER SERIES ∞

Let

∑a

m (x

− x0 ) m = a0 + a1 ( x − x0 ) + a2 ( x − x0 ) 2 + ......

...(1)

m=0

be a power series in x – x0 with centre x0 and real coefficients a0, a1, a2, ...... . *The elementary functions consists of algebraic functions, trigonometric functions, inverse trigonometric functions, exponential functions, logarithmic functions and all others that can be constructed from these by adding, subtracting, multiplying, dividing or forming a function of a function. For exam3

ple, sin x + tan x2, ecot x +

log x ,

x + x 2 e x + sec x log (1 + x 2 )

3

are elementary functions.

4

DIFFERENTIAL EQUATIONS AND INTEGRAL TRANSFORMS

This power series is said to be convergent at x = x1, if n

lim

n→∞

∑a

m ( x1

− x0 ) m

m=0

exists finitely and in this case the sum of the power series

∞

∑ am ( x1 − x0 )m

is the value of

m=0

this limit.

The power series (1) is always convergent at x = x0 because in this case the power series (1) becomes a0 + 0 + 0 + ...... If there are other values of x for which (1) is convergent, then these values form an interval, called the convergence interval. If the convergence interval is finite then it is of the form | x – x0 | < R. The constant R is called the radius of convergence of (1). ∞

The radius of convergence of the series

1 lim

m→∞

m (x

− x0 ) m can be found by using either of

m=0

the following formulae : (i) R =

∑a

(ii) R =

m |a | m

lim

m→∞

1 am + 1

.

am

In this case, the above limits are non-zero finite numbers. In case, lim

m→∞

m| a | m

F or GH

lim

m→∞

am + 1 am

I is zero, then the power series (1) converges for JK

all x and its convergence interval is infinite and radius of convergence, R = ∞. If the power series (1) converges only at its centre x0, then its radius of convergence, R is defined to be zero.

Thus, we see that radius of convergence exists for each type of power series. If the radius of convergence R is finite and non-zero, then the power series is convergent inside the interval |x – x0| < R that is, x0 – R < x < x0 + R and divergent outside this interval. A power series may or may not converge at the end points of its interval of convergence. 1.4. OPERATIONS ON POWER SERIES (i) Termwise differentiation. Let

∞

∑ am ( x − x 0 ) m

m=0

be a power series in x – x0 with

centre x0 and real coefficients a0, a1, a2, ...... . Let R (> 0) be the radius of convergence of this power series. ∴ This power series converges in the interval of convergence | x – x0 | < R. Let its sum be deonoted by f(x). ∴

f(x) =

∞

∑ am ( x − x0 )m = a0 + a1( x − x0 ) + a2 ( x − x0 )2 + ... for|x − x0 |< R .

m=0

It can be proved that the power series on the right side can be differentiated term by term and we have