Applied Mathematics III
Page 1: Applied Mathematics III Exam Instructions
Institution: Adarsh Management Institute of India
Certification: ISO 9001:2008 Certified International B-School
Subject: Applied Mathematics III
Total Marks: 100
Instructions
Attempt all questions.
Make suitable assumptions wherever necessary.
Figures to the right indicate full marks.
Questions
Q.1 - Do as directed (Total: 15 marks)
(a) Solve:
Equation: dy + y = x.dx
(b) Evaluate:
Integral: ∫exp(-x²)dx from 0 to ∞.
(c) Find:
L{sin 2t cos 2t}
(d) State:
The generating function and integral representation for the Bessel function Jn(x).
(e) Prove that:
2sin(x) * J₃(x) = -cos(x) / (2πx).
(f) Show that:
x³ = 2P(x) + 3P₂(x).
(g) Find:
The Fourier transform of the functionf(x) = { k, 0 < x < a; 0, otherwise}
Q.2 (Total: 25 marks)
(a) Using Laplace transform:
Solve the initial value problem:y'' + 2y' + y = e^(-t), y(0) = -1, y'(0) = 1.
(b) Solve the following differential equations (Any 2):
2xy dx + x² dy = 0
dy/dx - y = e^(2x)
x dy/dx + y = -1
OR
(b) (i) Using beta and gamma functions:
Simplify: B(m, n) B(m+n, p) B(m+n+p, q)
(ii) Express:
∫(x^m) (1 - x^n)^p dx in terms of Gamma function.
(iii) State:
Legendre duplication formula.
(iv) Prove that:
B(m,m) B(m+1,m+1) = π^(m-1)/(2*2^(4m)).
Page 2: Continuation of Questions
Q.3 (Total: 20 marks)
(a) Solve the initial value problem:
y'' + y' - 2y = 0, y(0) = 4, y'(0) = -5
(b) Given functions:
e^x and e^(-x), on interval [a, b].
Are these functions linearly independent or dependent?
(c) Using variation of parameters:
Solve the differential equation: y'' + y = sec(x).
OR
Q.3 (Continuation) (Total: 20 marks)
(a) Prove that:
d[x^(n+1)J(n+1)(x)]/dx = x^(n+1)J(n)(x).
(b) Attempt any three from the following:
Express polynomial x³ + 2x² - x - 3 in terms of Legendre polynomials.
Show that ∫ Pm(x)Pn(x)dx = 0, if m ≠ n.
Evaluate Pn(-1) using generating relation of Legendre polynomials.
Obtain the value of ∫ n².
Page 3: Further Questions
Q.4 (Total: 20 marks)
(a) Find the Fourier series for:
The function f(x) = x² for -π < x < π.
(b) Obtain the Fourier series for periodic function:
f(x) = 2x for -1 < x < 2, period: p = 2L = 2.
(c) Find:
Fourier transform of the function exp(-ax²).
OR
(a) Using undetermined coefficients method:
Solve y'' + 4y = 8x².
(b) Using series solution method:
Solve y'' + y = 0.
(c) Find:
The steady state oscillation of the mass-spring system governed by y'' + 3y' + 2y = 20cos(2t).
Q.5 (Total: 20 marks)
(a) Attempt (any two):
Evaluate: L^(-1)[1 / ((s + 2)(s - 3))].
Evaluate: L^(-1)(s² + 6s + 18).
First Shifting Theorem:
Obtain: L{(t + 1)²e^t}.
(b) Find the value of:
L{t sin(ωt)}.
Convolution product 1 * 1 where * denotes convolution product.
OR
(a) Find solution u(x,y) of the PDE u_xx + u_yy = 0
via the method of separation of variables.
(b) Attempt (any one):
Prove Laplacian u in polar coordinates: ∇²u = (1/r)(∂u/∂r) + (1/r²)(∂²u/∂θ²).
Find the potential inside a spherical capacitor with radius 1 ft: upper hemisphere at 110 volts, lower grounded.