GTU Advanced Engineering Mathematics - June 2014 Exam Question Paper

GTU Computer Engineering (Semester 3)
Advanced Engineering Mathematics
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'=e^2x+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:

\frac{s}{s^{2} - 3 s + 2}

$\dfrac{s}{s^{2}-3s+2}$

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

Γ (\frac{3}{4} | \frac{1}{4})

$\Gamma \left ( \dfrac34\Big|\dfrac14 \right )$

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} x^{2} e^{- x^{4}} d x

$\int_{0}^{\infty }\limits x^{2}e^{-x^{4}}dx$

2 M

2 (a) (ii) Solve: (D^4>-1)y=0.

2 M

2 (a) (iii) Solve the partial differential equation u_xy=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) = \frac{s^{2}}{(s^{2} + 4) (s^{2} + 9)}

$f(t)=\dfrac{s^{2}}{(s^{2}+4)(s^{2}+9)}$

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

$\int_{-1}^{1}\limits \left ( 1-x^{2} \right )}^{n}dx$ where n is a positive integer.
ii)

\int_{0}^{π / 2} \sqrt{\sin θ d θ} \times \int_{0}^{π / 2} \frac{1}{\sqrt{\sin θ}} d θ

$\int_{0}^{\pi/2}\limits \sqrt{\sin \theta d \theta}\times \int_{0}^{\pi/2}\limits \dfrac{1}{\sqrt{\sin \theta}}d \theta$

7 M

4 (a) i) Prove that

J_{\frac{3}{2}} (x) = \sqrt{\frac{2}{π x}} [\frac{\sin x}{x} - \cos x]

$J_{\dfrac{3}{2}}(x)=\sqrt{\dfrac{2}{\pi x}}\left [ \dfrac{\sin x}{x}- \cos x \right ]$
ii) p_n(-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

x d y - y d x = \sqrt{x^{2} + y^{2}}

$xdy-ydx=\sqrt{x^{2}+y^{2}}$
ii) Solve:

x^{3} \frac{d^{3} y}{d x^{3}} + 2 x^{2} \frac{d^{2} y}{d x^{2}} + 2 y = 10 (x + \frac{1}{x})

$x^{3}\dfrac{d^{3}y}{dx^{3}}+2x^{2}\dfrac{d^{2}y}{dx^{2}}+2y=10\left ( x+\dfrac{1}{x} \right )$ .

7 M

4 (d) i) If y_{1=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)=e^x 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} (x \log x)^{3} d x

$\int_{0}^{1}\limits (x \log x)^{3}dx$ .
ii) By using the relation between Beta and Gamma function prove that

β (m, n) β (m + n, p) β (m + n + p, q) = \frac{Γ m Γ n Γ p Γ q}{Γ (m + n + p + q)}

$\beta (m,n)\beta(m+n,p)\beta(m+n+p,q)=\dfrac{\Gamma m\Gamma n\Gamma p\Gamma q}{\Gamma (m+n+p+q)}$

7 M

5 (c) Solve Completely the equation

\frac{\partial^{2} y}{\partial x^{2}} = c^{2} \frac{\partial^{2} y}{\partial x^{2}}

$\dfrac{\partial^2 y }{\partial x^2}=c^{2}\dfrac{\partial^2 y }{\partial x^2}$ 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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