GTU Computer Engineering (Semester 3)
June 2014
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) (i) Find the general solution of the differential equation y'=e2x+3y
2 M
1 (a) (ii) Find the particular solution of the differential equation y?+4y=2sin3x by using method of undetermined coefficients.
2 M
1 (a) (iii) Find the inverse Laplace transform of following function: $\dfrac{s}{s^{2}-3s+2}$
3 M
1 (b) (i) i) Define Ordinary Point of the differential equation y''+P(x)y'+Q(x)y=0
2 M
1 (b) (ii) Find the value of $\Gamma \left ( \dfrac34\Big|\dfrac14 \right )$
2 M
1 (b) (iii) Express the function f(x)= x as a Fourier series in interval[-π,π]
3 M

2 (a) (i) i) Evaluate $\int_{0}^{\infty }\limits x^{2}e^{-x^{4}}dx$
2 M
2 (a) (ii) Solve: (D4>-1)y=0.
2 M
2 (a) (iii) Solve the partial differential equation uxy=x3+y3.
3 M
2 (b) (i) Find the Laplace transforms of function f(t)=t5+cos5t+e-100t.
3 M
2 (b) (ii) Using method of variation of parameters solve the differential equation
y?+4y=tan2x.
4 M

3 (a) Find the Laplace transforms of following functions: (i) cos3 t (ii) sin2 t .
7 M
3 (b) State Convolution Theorem and using it find inverse Laplace transform of function $f(t)=\dfrac{s^{2}}{(s^{2}+4)(s^{2}+9)}$
7 M
3 (c) Using Laplace transform solve the differential equation:
y''+5y'+6y=e-2,y(0)=0,y'(0)=-1.
7 M
3 (d) Evaluate i) $\int_{-1}^{1}\limits \left ( 1-x^{2} \right )}^{n}dx$ where n is a positive integer.
ii) $\int_{0}^{\pi/2}\limits \sqrt{\sin \theta d \theta}\times \int_{0}^{\pi/2}\limits \dfrac{1}{\sqrt{\sin \theta}}d \theta$
7 M

4 (a) i) Prove that $J_{\dfrac{3}{2}}(x)=\sqrt{\dfrac{2}{\pi x}}\left [ \dfrac{\sin x}{x}- \cos x \right ]$
ii) pn(-1)n=(-1)n.
7 M
4 (b) (i) Solve the differential equation y?+xy=0 by the power series method.
ii) State Rodrigue's Formula and using it compute P0(x),P1(x).
7 M
4 (c) i) Solve the differential equation $xdy-ydx=\sqrt{x^{2}+y^{2}}$
ii) Solve: $x^{3}\dfrac{d^{3}y}{dx^{3}}+2x^{2}\dfrac{d^{2}y}{dx^{2}}+2y=10\left ( x+\dfrac{1}{x} \right )$.
7 M
4 (d) i) If y1=x is one solution of x2 y+xy-y=0 then find the second solution.
Solve :$(2x+5)^{2}\dfrac{d^{2}y}{dx^{2}}-6(2x+5)\dfrac{dy}{dx}+8y=6x$.
7 M

5 (a) (i) Find half range cosine series for the function f(x)=ex in interval [0,2].
(ii) Express the function f(x)=x-x2 as a Fourier series in interval [-ππ].
7 M
5 (b) i) Evaluate $\int_{0}^{1}\limits (x \log x)^{3}dx$.
ii) By using the relation between Beta and Gamma function prove that
$\beta (m,n)\beta(m+n,p)\beta(m+n+p,q)=\dfrac{\Gamma m\Gamma n\Gamma p\Gamma q}{\Gamma (m+n+p+q)}$
7 M
5 (c) Solve Completely the equation $\dfrac{\partial^2 y }{\partial x^2}=c^{2}\dfrac{\partial^2 y }{\partial x^2}$ representing the vibrations of a string of length l fixed at both ends given that,
$y(0,t)=y(l,t)=0,y(x,0)=f(x),\dfrac{\partial y}{\partial t}(x,0)=0,0 7 M 5 (d) Find the Fourier Transform of the function f defined as follows: \[f(x)=\begin{matrix} x &\left | x \right | &a. \end{matrix}$
7 M

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