In this undergraduate/graduate textbook, the authors introduce ODEs and PDEs through 50 class-tested lectures. Mathematical concepts are explained with clarity and rigor, using fully worked-out examples and helpful illustrations. Exercises are provided at the end of each chapter for practice. The treatment of ODEs is developed in conjunction with PDEs and is aimed mainly towards applications. The book covers important applications-oriented topics such as solutions of ODEs in form of power series, special functions, Bessel functions, hypergeometric functions, orthogonal functions and polynomials, Legendre, Chebyshev, Hermite, and Laguerre polynomials, theory of Fourier series.

Undergraduate and graduate students in mathematics, physics and engineering will benefit from this book. The book assumes familiarity with calculus.

0 x2 y ′′ + 2xy ′ + xy = 0, x = 0 x2 y ′′ + 4xy ′ + (2 + x)y = 0, x = 0 x(1 − x)y ′′ − 3xy ′ − y = 0, x = 0 x2 y ′′ − (x + 2)y = 0, x = 0 x(1 + x)y ′′ + (x + 5)y ′ − 4y = 0, x = 0 (x − x2 )y ′′ − 3y ′ + 2y = 0, x = 0. 6.3 A supply of hot air can be obtained by passing the air through a heated cylindrical tube. It can be shown that the temperature T of the air in the tube satisfies the diﬀerential equation d2 T 2πrh upC dT + (Tw − T ) = 0, − (6.10) 2 dx kA dx kA where x = distance from intake end

distribution of rectangular and circular plates in the transient state. Again using the method of separation of variables, in Lecture 37 we find vertical displacements of thin membranes occupying rectangular and circular regions. The three-dimensional Laplace equation occurs in problems such as gravitation, steady-state temperature, electrostatic potential, magnetostatics, fluid flow, and so on. In Lecture 38 we find the Fourier series solution of the Laplace equation in a three-dimensional box

Qn−1 (x), where Qn−1 (x) is a n+1 polynomial of degree at most n−1 (ii) Use x2 Pn−1 (x) = 2n−1(2n−2)! ((n−1)!)2 x +Qn (x) and (i) (iii) Use (7.13) (iv) Use Problem 7.8(i) and then Problem 7.8(iii) (v) Use Corollary 13.2 and (7.9) (vi) Integrate by parts and use (3.19) with a = n (vii) Use Problem 7.8(iv) and (7.13). 13.7. From (7.8), Pn (x) = (2n)! 2n (n!)2 (x − x1 ) · · · (x − xn ). Orthogonal Functions and Polynomials (Cont’d.) 1 1 π (1 − x2 )−1/2 Π2 (x)dx = 22n−1 2n−1 Tn (x), −1 1 1 π 2

(β) = 0 y(0) = 0, y(β) + y ′ (β) = 0 y(0) − y ′ (0) = 0, y ′ (β) = 0 152 Lecture 19 (vi) y(0) − y ′ (0) = 0, y(β) + y ′ (β) = 0. 19.3. Find the eigenvalues and eigenfunctions of each of the following Sturm–Liouville problems: (i) (ii) (iii) (iv) (v) (vi) y ′′ + λy = 0, y(0) = y(π/2) = 0 y ′′ + (1 + λ)y = 0, y(0) = y(π) = 0 y ′′ + 2y ′ + (1 − λ)y = 0, y(0) = y(1) = 0 (x2 y ′ )′ + λx−2 y = 0, y(1) = y(2) = 0 x2 y ′′ + xy ′ + (λx2 − (1/4))y = 0, y(π/2) = y(3π/2) = 0 ((x2 + 1)y ′ )′ + λ(x2 +

= 0 is a singular point, but every other point is an ordinary point. A singular point x0 at which the functions p(x) = (x − x0 )p1 (x) and q(x) = (x − x0 )2 p2 (x) are analytic is called a regular singular point of the DE (2.1). Thus, a second-order DE with a regular singular point x0 has the form p(x) q(x) y ′′ + y′ + y = 0, (3.10) (x − x0 ) (x − x0 )2 where the functions p(x) and q(x) are analytic at x = x0 . Hence, in Example 3.3 the point x0 = 0 is a regular singular point. If a singular