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

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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.

No.	Topic	Concept Summary
(a)	Second-Order Differential Equation	Solve d2ydx2=−12x2+26x−10\frac{d^2y}{dx^2} = -12x^2 + 26x - 10dx2d2y=−12x2+26x−10 using given conditions x=0,y=5x=0, y=5x=0,y=5 and x=0,y′=21x=0, y'=21x=0,y′=21; find yyy at x=1x=1x=1.
(b)	Equal Roots in Characteristic Equation	For d2ydx2+2αdydx+y=0\frac{d^2y}{dx^2} + 2\alpha \frac{dy}{dx} + y = 0dx2d2y+2αdxdy+y=0, find α when roots are equal → α=1\alpha = 1α=1.
(c)	Legendre Polynomial Property	Prove Pn(1)=1P_n(1) = 1Pn(1)=1 and Pn(−1)=(−1)nP_n(-1) = (-1)^nPn(−1)=(−1)n.
(d)	Bessel Function Property	Show J−n(x)=(−1)nJn(x)J_{-n}(x) = (-1)^n J_n(x)J−n(x)=(−1)nJn(x) and J−n(−x)=(−1)nJn(x)J_{-n}(-x) = (-1)^n J_n(x)J−n(−x)=(−1)nJn(x).
(e)	Laplace Transform of an Integral	L{∫0tf(τ)dτ}=F(s)sL \left\{ \int_0^t f(\tau) d\tau \right\} = \frac{F(s)}{s}L{∫0tf(τ)dτ}=sF(s).
(f)	Laplace of te−2tcos⁡tt e^{-2t} \cos tte−2tcost	Standard transform identity involving shift theorem.
(g)	Fourier Coefficients for f(x)=x2f(x) = x^2f(x)=x2	For 0<x<2π0 < x < 2\pi0<x<2π: a0=2π23,an=−4n2,bn=0a_0 = \frac{2\pi^2}{3}, a_n = -\frac{4}{n^2}, b_n = 0a0=32π2,an=−n24,bn=0.
(h)	PDE of Spheres with Centers on Z-Axis	z2(1+p2+q2)=a2z^2(1 + p^2 + q^2) = a^2z2(1+p2+q2)=a2 where p=∂z∂x,q=∂z∂yp = \frac{\partial z}{\partial x}, q = \frac{\partial z}{\partial y}p=∂x∂z,q=∂y∂z.
(i)	Eliminate Arbitrary Function	Eliminate ϕ from ϕ(x2+y2+2z+2y,z)=0ϕ(x^2 + y^2 + 2z + 2y, z) = 0ϕ(x2+y2+2z+2y,z)=0 to get PDE by differentiation.
(j)	Classification of PDEs	Based on B2−4ACB^2 - 4ACB2−4AC: Elliptic (negative), Parabolic (zero), Hyperbolic (positive).

SECTION – B (3 × 7 = 21 Marks)

Medium-level problems involving derivations and applications.

Question	Concept	Key Idea
(a)	Exponential Decay Equation	Solve d2n(x)dx2−n(x)L2=0\frac{d^2n(x)}{dx^2} - \frac{n(x)}{L^2} = 0dx2d2n(x)−L2n(x)=0 with n(0)=x,n(∞)=0n(0)=x, n(\infty)=0n(0)=x,n(∞)=0. Solution → n(x)=xe−x/Ln(x) = x e^{-x/L}n(x)=xe−x/L.
(b)	Frobenius Method	Series solution of (1−x2)y′′−2xy′+2y=0(1-x^2)y'' - 2xy' + 2y = 0(1−x2)y′′−2xy′+2y=0 around x=0x=0x=0.
(c)	Damped Mass-Spring System (Laplace Transform)	y′′+5y′+6y=f(t)y'' + 5y' + 6y = f(t)y′′+5y′+6y=f(t) where f(t)f(t)f(t) is a square wave; solved via Laplace techniques.
(d)	Fourier Cosine Series of f(x)=xsin⁡xf(x) = x\sin xf(x)=xsinx	Expansion over (0,π)(0, \pi)(0,π), used to prove series identity 1/(1×3)−1/(3×5)+…=(π−2)/41/(1×3) - 1/(3×5) + … = (\pi - 2)/41/(1×3)−1/(3×5)+…=(π−2)/4.
(e)	PDE by Separation of Variables	Solve x∂u∂x+2∂u∂y=0,u(x,0)=3e−xx \frac{\partial u}{\partial x} + 2 \frac{\partial u}{\partial y} = 0, u(x,0) = 3e^{-x}x∂x∂u+2∂y∂u=0,u(x,0)=3e−x. Gives u(x,y)=3e−xu(x,y) = 3e^{-x}u(x,y)=3e−x.

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}.y1dxdy2−y2dxdy1=Ce−∫Pdx.

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)2πxJ3/2(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)∫−11Pm(x)Pn(x)dx=0(m=n)

∫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+1nPn−1(x)+2n+1n+1Pn+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=cos2tx(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∑∞n21cos(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

Unit	Topics Covered	Techniques Used
I	Ordinary Differential Equations	Cauchy-Euler, Frobenius, Variation of Parameters
II	Laplace Transform & Applications	Convolution, Periodic Functions, System Response
III	Fourier Series & Integrals	Cosine/Sine Series, Parseval’s theorem
IV	Special Functions	Bessel, Legendre, Orthogonality, Recurrence
V	PDEs	Separation of Variables, Classification, Heat & Wave Equations