GTU Advanced Engineering Mathematics - December 2016 Exam Question Paper

GTU Computer Engineering (Semester 3)
Advanced Engineering Mathematics
December 2016

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(10) Give the differential equation of the orthogonal trajectory of the family of circles

x^{2} + y^{2} = a^{2} .

$x^2+y^2=a^2.$

1 M

1(11) Find the Wronskian of the two function sin2x and cos2x.

1 M

1(12) Solve

(D^{2} + 6 D + 9) x = 0; D = \frac{d}{d t} .

$\left ( D^2+6D+9 \right )x=0; D =\frac{d}{dt}.$

1 M

1(13) To solve heat equation

\frac{\partial u}{\partial t} = C^{2} \frac{\partial^{2} u}{\partial x^{2}}

$\frac{\partial u}{\partial t}=C^2\frac{\partial^2u }{\partial x^2}$ / how many initial and boundary conditions are required.

1 M

1(14) From the partial differential equations from z = (x+at) +g(x-at).

1 M

1(2) State relation between beta and gamma function.

1 M

1(3) Define Heaviside's unit step function.

1 M

1(4) Define Laplace transform of f(t),t≥0.

1 M

1(5) Find Laplace transform of

t^{\frac{- 1}{2}} .

$t^{\frac{-1}{2}}.$

1 M

1(6) Find

L {\frac{S i n a t}{t}}, given that L {\frac{S i n t}{t}} = {tan}^{- 1} {\frac{1}{s}} .

$L\left \{ \frac{Sinat}{t} \right \},\ \ \text{given that L}\left \{ \frac{Sint}{t} \right \} = \tan ^{-1}\left \{ \frac{1}{s} \right \}.$ /

1 M

1(7) Find the continuous extension of the function

f (x) = \frac{x^{2} + x - 2}{x^{2} - 1} to x = 1

$f\left ( x \right )=\frac{x^2+x-2}{x^2-1}\ \text{to}\ \ x=1$

1 M

1(8) Is the function

f (x) = \frac{1}{x}

$f\left ( x \right )=\frac{1}{x}$ / continous on [-1, 1]? Give reason.

1 M

1(9) Solve

\frac{d y}{d x} = e^{3 x - 2 y} + x^{2} e^{- 2 y} .

$\frac{dy}{dx}=e^{3x-2y}+x^2e^{-2y}.$

1 M

1(a) Find ⌈

(\frac{1}{2}) .

$\left ( \frac{1}{2} \right ).$

1 M

2(a) Solve:

(x + 1) \frac{d y}{d x} - y = e^{3 x} (x + 1) .

$\left ( x+1 \right )\frac{dy}{dx}-y = e^{3x}(x+1).$

3 M

2(b) Solve:

\frac{d y}{d x} + \frac{y cos x + sin y + y}{sin x + x cos y + x} = 0

$\frac{dy}{dx}+\frac{y\cos x+\sin y+y}{\sin x+x\cos y+x}=0$

4 M

Solve any one question.Q2(c) &Q2(d)

2(c) Fin the series solution of

\frac{d^{2} y}{d x^{2}} + x y = 0.

$\frac{d^2y}{dx^2}+xy=0.$

7 M

2(d) Find the general solution of

2 x^{2} y^{''} + x y^{'} + (x^{2} - 1) y = 0

$2x^2y''+xy'+\left ( x^2-1 \right )y=0$ / by using frobenius method.

7 M

solve any one question Q.3(a,b,c) &Q4(a,b,c)

3(a) Solve :

(D^{3} - 3 D^{2} + 9 D - 27) y = cos 3 x .

$\left ( D^3-3D^2+9D-27 \right )y=\cos 3x.$

3 M

3(b) Solve :

x^{2} \frac{d^{2} y}{d x^{2}} + 4 x \frac{d y}{d x} + 2 y = x^{2} sin (l n x) .

$x^2\frac{d^2y}{dx^2}+4x\frac{dy}{dx}+2y=x^2\sin \left ( lnx \right ).$

4 M

3(c)(i) Solve :

\frac{\partial^{3} z}{\partial x^{3}} - 2 \frac{\partial^{3} z}{\partial x^{2} \partial y} = 2 e^{2 x} .

$\frac{\partial ^3z}{\partial x^3}-2\frac{\partial^3z }{\partial x^2 \partial{y}}=2e^{2x}.$

3 M

3(c)(ii) Find the general solutiion to the partial differential equation

(x^{2} - y^{2} - z^{2}) + 2 x y q = 2 x z .

$\left ( x^2-y^2-z^2 \right )+2xyq=2xz.$

4 M

4(a) Solve :

(D^{3} - D) = x^{3} .

$\left ( D^3-D \right )=x^3.$

3 M

4(b) Find the solution

y^{''} - 3 y^{'} + 2 y = e^{x},

$y''-3y'+2y=e^x,$ / using the method of variation of parameters.

4 M

4(c) Solve

x \frac{\partial u}{\partial x} - 2 y \frac{\partial u}{\partial y} = 0

$x\frac{\partial u}{\partial x}-2y\frac{\partial u}{\partial y}=0$ / using method of separation of variables.

7 M

solve any one question Q.5(a,b,c) &Q6(a,b,c)

5(a) Find the Fourier cosine integral of

f (x) = e^{k x}, x > 0, k > 0

$f\left ( x \right )= e^{kx},x>0, k>0$

3 M

5(b) Express \( f(x)=|x|,-\pi / as fouries series.

4 M

5(c) Find Fourier Series for the function f(x) given by

$\left\{\begin{matrix} 1+\frac{2x}{\pi }; &-\pi \leq x\leq 0 \\ 1-\frac{2x}{\pi } & 0\leq x\leq \pi \end{matrix}\right.$ / Hence deduce that

$\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+....\frac{\pi ^2}{8}.$

7 M

6(a) Obtain the Fourier Series of periodic function function \[f(x)=2x,-1

3 M

6(b) Show that \[

4 M

6(c) Expand f(x) in Fourier series in the interval (0,2π) if \( f(x)\left\{\begin{matrix} -\pi ; &0,/ and hence show that

$\sum_{r=0 }^{\infty }\frac{1}{\left ( 2r+1 \right )^2}=\frac{\pi ^2}{8}.$

7 M

solve any one question Q.7(a,b,c) &Q8(a,b,c)

7(a) Find L {{\int ^t_0 e^t\frac{\sin t}{t}dt}}

3 M

7(b) Find

$L^{-1}\left \{ \frac{2s^2-1}{\left ( s^2+1 \right )\left ( s^2 +4 \right )} \right \}$

4 M

7(c) Solve initial value problem :

$y''-3y'+2y=4t+e^{3t},y(0)=1$ / and y'(0)=-1, using Laplace transform.

7 M

8(a) Find

$L\left \{ t\sin 3t\cos 2t \right \}.$

3 M

8(b) Find

$L^{-1}\left \{ \frac{e^{-35}}{s^2+8s+25} \right \}.$

4 M

8(c) State the convolution theorem and apply it to evaluate

$L^{-1}\left \{ \frac{s}{\left ( s^2+a^z \right )^z} \right \}.$

7 M

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