This paper is about problem solving course on differential equations. It would invite the students to the subject of Mathematics and would inspire them to look to some comprehensive books on problem solving. This paper consists of several examples of differential equations and is based on lectures given to final year students of master programme. All examples are completely given in a lucid student friendly manner. A large number of solved exercises are included.
Table of Contents
Q-1 Solve y''+xy'+y = 0 near 0.
Q-2 Solve (1 - x2)y'' - 2xy' + n(n + 1)y = 0 near 0 ; where n is constant.
Q-3 State and solve Bessel’s differential equation of order P near 0.
Q-4 Check the nature of ∞ for x4y'' + x3(x + 2)y' + y = 0
Q-5 Find an ordinary and singular points of the following differential equation
Q-6 Discuss the behavior of singular soution of Bessel’s equation and find its indicial equation.
Q-7 Derive an integral representation for the Gauss Hyper geometric function.
Q-8 Define regular singular point. Find Frobenius series expansion 2x2y'' + x(2x + 1)y' - y = 0 near regular singular point.
Q-9 State only Rodrigue’s formula and find Legendre polynomials P0(x),P1(x),P2(x) and P3(x). Also express x3 + 5x + 2 in terms of Legendre polynomials.
Q-10 State and prove orthogonality of Legendre polynomials.
Q-11 State and prove orthogonality of Bessel’s function.
Q-12 (I) Prove that (n + 1)Pn+1 = (2n + 1)xPn - nPn-1
Q-13 Prove that d/dx [xnJn(x)] = xnJn-1(x)
Q-14 Prove that d/dx [x-nJn(x)] = -x-nJn+1(x)
Q-15 State and prove Rodrigue’s formula for the Legendre polynomial and hence express x2-3x+1 in terms of Legendre polynomials.
Q-16 In usual notation prove that integral -1 to 1 Pm(x)Pn(x)dx
Q-17 Prove that nPn = (2n - 1)xPn-1 - (n - 1)Pn-2; n ≥ 2
Q-18 Prove that Jn(x+y) = summation k=-∞ to ∞ Jn-k(x)Jk(y) and hence deduce that 1+J0^2(x)+2 summation k=1 to ∞ Jk^2(x)
Q-19 Show that all the roots of Pn(x) are real and lies between -1 and 1.
Q-20 Prove that d/dx F(α, β; γ; x) = αβ/γ F(α + 1, β + 1; γ + 1; x).
Q-21 Prove that integral x3J0(x)dx = x3J1(x) - 2x2J2(x) + c
Q-22 Prove that Jm(x) = 1/π integral 0 to π cos(mθ-xsinθ)dθ and hence deduce that J0(x) = 2/π integral 0 to π/2 cos(xsinθ)dθ
Q-23 Find the Fourier Legendre expansion of a function defined by f(x)
Q-24 Find the third approximation of the solution of the equation dy/dx = z, dz/dx = x2z + x4y by Picard’s method,y = 5 and z = 1 when x = 0.
Q-25 Solve by Picard’s method
Q-26 Find three successive approximations of the solution of dy/dx = ex + y2, y(0) = 0
Q-27 Determine which of the following equations are integrable and find the solution of those which are integrable
Q-28 Prove that d/dx [x-pJp(x)] = -x-pJp+1(x); n ≥ 1 and hence show that
Q-29 Prove that
Q-30 Prove that
Q-31 Solve p2x + q2 = z by using Jacobi’s method.
Q-32 Solve 2(z + xp + yq) = yp2 by using Charpit’s method.
Q-33 State and prove necessary and sufficient condition for compatibility of partial differential equation f(x, y, z, p, q) = 0 and g(x, y, z, p, q) = 0.
Q-34 Solve yzp + zxq = xy by Langrange’s method.
Q-35 Solve
Q-36 Find complete integral of q = 3p2
Q-37 Solve z = px + qy + p2 + q2 by using Charpit’s method.
Q-38 Find the general solution of
Q-39 Classify the equation and convert it into canonical form : 4r - y6t = 3y5q
Q-40 Convert the equation into canonical form : r + 2s + t = 0
Q-41 Using Monge’s method , solve the equation : r + s - 6t = 0
Q-42 Using Monge’s method , solve the equation : rt - s2 + 1 = 0
Q-43 By separating the variables , find the solution of three dimensional Laplace equation in cylindrical coordinate system.
Q-44 Solve wave equation in cartesian coordinates by method of separation variable and show that solution is ψ(x, y, z, t) = e±i(lx+my+nz+kct)
Q-45 If (βD' + γ)2 is a factor of F(D, D') ,then e-γ/β y [φ1(βX) + yφ2(βX)] is a solution of F(D, D')z = 0
Objectives and Topics
This work systematically addresses advanced analytical methods for solving various classes of differential equations, including ordinary differential equations (ODEs), Bessel's and Legendre's equations, and partial differential equations (PDEs) through separation of variables and Monge's or Charpit's methods.
- Power series solutions for differential equations near ordinary and singular points.
- Properties and orthogonality of special functions (Bessel and Legendre).
- Exact and approximate solutions for first-order nonlinear PDEs.
- Canonical forms and classification of second-order linear PDEs.
- Analytical representation of mathematical functions through integral transforms and series expansions.
Excerpt from the Book
Q-1 Solve y''+xy'+y = 0 near 0.
Given that y'' + xy' + y = 0 ...........................(1) Comparing (1) with y'' + P(x)y' + Q(x)y = 0 , Here P(x) = x, Q(x) = 1 Since P(x) = x and Q(x) = 1 both are analytic at x = 0 is an ordinary point of (1)
To solve (1) , we take y = c0+c1x+c2x2+c3x3+..............= summation n=0 to ∞ cnxn .............................................(2)
Differentiating (2) twice in succession with respect to ’x’ , we get y' = summation n=1 to ∞ cnnxn-1 and y'' = summation n=1 to ∞ cnn(n - 1)xn-2 ............................................(3) Putting the above values of y, y' and y'' in (1) summation n=1 to ∞ cnn(n - 1)xn-2 + x summation n=1 to ∞ cnnxn-1 + summation n=0 to ∞ cnxn = 0 or summation n=1 to ∞ cnn(n - 1)xn-2 + summation n=1 to ∞ cnnxn + summation n=0 to ∞ cnxn = 0 or summation n=1 to ∞ cn+2(n + 2)(n + 1)xn + summation n=1 to ∞ cnnxn + summation n=0 to ∞ cnxn = 0 or [2c2 + c0]+summation n=1 to ∞ [cn+2(n+2)(n+1)+n cn + cn]xn = 0 ............................(4) Since (4) is an identity , equating the constant term and the coefficient of various powers of x to zero , we get 2c2 + c0 = 0 ⇒ c2 = -c0/2 .......................(5) and cn+2(n + 2)(n + 1)+n cn + cn = 0 ; for all n ≥ 1 ....................(6) ⇒ cn+2 = - cn(n+1)/(n+1)(n+2) = - cn/(n+2) ............(7) Putting n = 1, 2, 3, ............ in (7) c3 = -c1/3 , c4 = -c2/4 = c0/8 , c5 = -c3/5 = c1/15 , c6 = -c4/6 = -c0/48 ...................... Putting these values in (1) , we get y = c0(1- x2/2 + x4/8 - x6/48 +..............) + c1(x- x3/3 + x5/15 -..............) ; c0 and c1 are arbitrary constants. Which is required solution.
Summary of Chapters
Q-1 to Q-8: These chapters focus on finding power series solutions for various linear differential equations near ordinary and regular singular points using Frobenius methods.
Q-9 to Q-18: This section covers the properties, orthogonality, and recurrence relations of Legendre polynomials and Bessel functions.
Q-19 to Q-23: These topics discuss mathematical proofs regarding the roots of polynomials, integral representations of functions, and Fourier-Legendre expansions.
Q-24 to Q-26: These chapters provide iterative approximation methods for solving differential equations, specifically employing Picard’s method.
Q-27 to Q-38: This part details the analytical solutions of first and second-order partial differential equations using methods like Jacobi’s, Charpit’s, and Lagrange’s techniques.
Q-39 to Q-45: These concluding sections deal with the classification, canonical forms of second-order PDEs, and method of separation of variables for wave and Laplace equations.
Keywords
Differential equations, Frobenius method, Bessel functions, Legendre polynomials, Picard’s method, partial differential equations, Charpit’s method, Monge’s method, orthogonality, series expansion, Laplace equation, wave equation, singular points, recurrence relations, analytical solutions.
Frequently Asked Questions
What is the fundamental focus of this publication?
The work focuses on providing analytical and numerical techniques to solve ordinary and partial differential equations, particularly those involving special mathematical functions.
Which specific special functions are analyzed?
The text extensively covers Bessel functions and Legendre polynomials, including their orthogonality and recurrence relations.
What is the primary objective of these mathematical methods?
The objective is to derive exact or approximate series solutions for equations that are often encountered in engineering and physics, such as wave or Laplace equations.
Which scientific methods are primarily utilized?
The publication utilizes Frobenius series expansions, Picard’s iteration method, Lagrange’s method for PDEs, Charpit’s and Jacobi’s methods for nonlinear PDEs, and the method of separation of variables.
What topics are discussed in the main body?
The main body treats the classification of PDEs, the conversion into canonical forms, and the integral representation of special functions.
How is the mathematical rigor maintained?
Rigor is maintained by consistently applying foundational theorems such as Rolle’s theorem and using rigorous proofs for orthogonality and compatibility conditions.
How is the nature of singular points determined in this text?
The text determines the nature of singular points by evaluating the analyticity of the coefficients P(x) and Q(x) in the standard form of the differential equation.
What significance do Legendre polynomials hold in the provided examples?
Legendre polynomials are shown to be useful for expanding complex polynomial expressions and solving boundary value problems through their orthogonality properties.
- Citar trabajo
- Nagjibhai Rabari (Autor), 2019, Solved Problems on Differential Equations, Múnich, GRIN Verlag, https://www.grin.com/document/499425
