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Roll No. ........................
Total No. of Questions: 05
Total No. of Printed Pages: 04
Total No. of Printed Pages: 04
AR-39
B.Sc.-B.Ed. (Reg./ATKT) Examination, 2024
(Third Semester)
MATHEMATICS
(3.2 Differential Equation)
Time: 3 Hours
Maximum Marks: 30
Note : All questions are compulsory. All question carry equal marks. Attempt any two parts from each questions.
(c)
Solve Recurrence relation for $J_n(x)$ :
(i)
$$ \frac{d}{dx} (x^n J_n(x)) = x^n J_{n-1}(x) $$
(ii)
$$ \frac{d}{dx} (x^{-n} J_n(x)) = -x^{-n} J_{n+1}(x) $$
1.
(a)
Define Power Series. Solve by power series method $y'' - y = 0$.
(b)
Express $f(x) = x^4 + 2x^3 - 2x^2 - 2$ in terms of Legendre's polynomials.
2.
(a)
Show that the function $1, \cos x, \sin x, \cos 2x, \sin 2x, \dots$ form an orthogonal set on an interval $-\pi \le x \le \pi$ and obtain orthonormal set.
(b)
Find the eigen values and Eigen functions of the Sturm-Liouville problem:
$$ \frac{d^2y}{dx^2} + \lambda y = 0 $$
(i)
$y(0) = 0$ and $y(\pi) = 0$.
(c)
Prove that:
$$ \int_{-1}^{1} P_n(x) dx = 0, \text{ if } n \ge 1 $$
3.
(a)
Find :
(i)
$L \{ \sin at \}$
(ii)
$L \{ \cos at \}$.
(b)
Find :
(i)
$L \{ e^t \sin^2 t \}$
(ii)
$L \{ e^t \cos^2 t \}$.
(c)
Find the Laplace transformation of the following functions:
(i)
$f(t) = t \sin^2 t$
(ii)
$f(t) = t \cos^2 t$.
4.
(a)
Construct a partial differential equation by eliminating arbitrary Constants $a$ if: $z = a(x + y)$
5.
(a)
Solve :
$$ \frac{\partial z}{\partial x} - \frac{\partial z}{\partial y} = x + y + z $$
(b)
Solve :
$$ p + r + s = 1 $$
(c)
Solve :
$$ \frac{\partial^2 z}{\partial x^2} - 2 \frac{\partial^2 z}{\partial x \partial y} + \frac{\partial^2 z}{\partial y^2} = \frac{1}{x^2} $$
(c)
Solve :
$$ \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} - 2z = e^{x-y} - x^2y^2 $$
