THEORY EXAMINATION (SEM–IV) 2016-17 MATHEMATICS-II

Course: B.Tech (All Engineering Branches)
Subject Code: RAS203
Subject Title: Mathematics–II
Exam Type: Theory (Semester IV, 2016–17)
Duration: 3 Hours
Maximum Marks: 70
Instruction: Be precise in answers; assume missing data if required.

SECTION – A (10 × 2 = 20 Marks)

Short and conceptual questions — testing formulas, properties, and basic problem-solving.

SECTION – B (3 × 7 = 21 Marks)

Medium-level problems involving derivations and applications.

SECTION – C (35 Marks Total)

Advanced, multi-step derivations and application-oriented mathematical problems.

Q3. Ordinary Differential Equations

(a) Solve d2ydx2+a2y=sec⁡(ax)\frac{d^2y}{dx^2} + a^2y = \sec(ax)dx2d2y​+a2y=sec(ax) using undetermined coefficients.
(b) Prove Wronskian identity:

y1dy2dx−y2dy1dx=Ce−∫Pdx.y_1 \frac{dy_2}{dx} - y_2 \frac{dy_1}{dx} = C e^{-\int P dx}.y1​dxdy2​​−y2​dxdy1​​=Ce−∫Pdx.

(c) Solve using variation of parameters:

y′′−6y′+9y=x2ex.y'' - 6y' + 9y = \frac{x^2}{e^x}.y′′−6y′+9y=exx2​. 

Q4. Special Functions

(a) Bessel Function Relation:

πx2J3/2(x)=1x(sin⁡x−xcos⁡x)\sqrt{\frac{\pi x}{2}} J_{3/2}(x) = \frac{1}{x}(\sin x - x \cos x)2πx​​J3/2​(x)=x1​(sinx−xcosx)

(b) Orthogonality of Legendre Polynomials:

∫−11Pm(x)Pn(x)dx=0(m≠n)\int_{-1}^{1} P_m(x) P_n(x) dx = 0 \quad (m \neq n)∫−11​Pm​(x)Pn​(x)dx=0(m=n)

(c) Prove recurrence relation:

∫xPn(x)dx=n2n+1Pn−1(x)+n+12n+1Pn+1(x)\int x P_n(x) dx = \frac{n}{2n+1} P_{n-1}(x) + \frac{n+1}{2n+1} P_{n+1}(x)∫xPn​(x)dx=2n+1n​Pn−1​(x)+2n+1n+1​Pn+1​(x) 

Q5. Laplace Transform Applications

(a) Laplace of Sawtooth Function:
For F(t)=Kt,0<t<1,period=1F(t) = Kt, 0 < t < 1, \text{period} = 1F(t)=Kt,0<t<1,period=1:

L{F(t)}=K(1−e−s)s2(1−e−s)L\{F(t)\} = \frac{K(1 - e^{-s})}{s^2(1 - e^{-s})}L{F(t)}=s2(1−e−s)K(1−e−s)​

(b) Convolution Theorem:
Find L−1{4s2+2s+5}L^{-1}\left\{\frac{4}{s^2 + 2s + 5}\right\}L−1{s2+2s+54​}.
(c) Simultaneous Differential Equations:
Using Laplace:

{x′=sin⁡t+yy′−2y=cos⁡2tx(0)=1,y(0)=0\begin{cases} x' = \sin t + y \\ y' - 2y = \cos 2t \end{cases} \quad x(0)=1, y(0)=0{x′=sint+yy′−2y=cos2t​x(0)=1,y(0)=0 

Q6. Fourier and PDE Problems

(a) Fourier Series Expansion:
For f(x)=(π−x2)2,0<x<2πf(x) = \left(\frac{\pi - x}{2}\right)^2, 0 < x < 2\pif(x)=(2π−x​)2,0<x<2π:

f(x)=π212−∑n=1∞1n2cos⁡(nx)f(x) = \frac{\pi^2}{12} - \sum_{n=1}^{\infty} \frac{1}{n^2} \cos(nx)f(x)=12π2​−n=1∑∞​n21​cos(nx)

(b) Operator Method PDE:
Solve (D′2+DΔ+2)u=sinh⁡x(D'^2 + DΔ + 2)u = \sinh x(D′2+DΔ+2)u=sinhx where symbols have usual meanings.
(c) First-Order PDE:
Solve p+5q=5z+tan⁡(y−5x)p + 5q = 5z + \tan(y - 5x)p+5q=5z+tan(y−5x).

Q7. Heat & Wave Equations

(a) Steady-State Temperature (2D Laplace Equation):
Square plate (0 ≤ x,y ≤ 20), boundaries at u=0u=0u=0 except top edge u(x,20)=x(20−x)u(x,20) = x(20-x)u(x,20)=x(20−x).
Solution → via separation of variables and Fourier sine series.

(b) One-Dimensional Heat Conduction (Transient State):
Bar (10 cm) with ends at 200°C and 300°C initially, then changed to 500°C and 100°C.
Find temperature distribution u(x,t)u(x,t)u(x,t) using Fourier series method.

Key Topics Summary