GTU Advanced Engineering Maths - May 2015 Exam Question Paper

GTU Information Technology (Semester 3)
Advanced Engineering Maths
May 2015

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) Solve the differential equation

\frac{d y}{d x} + \frac{1}{x} = \frac{e^{y}}{x^{2}}

$\dfrac {dy}{dx}+\dfrac {1}{x} = \dfrac {e^y}{x^2}$

4 M

1 (a) (ii) Solve the differential equation ye^x dx+(2y+e^x)dy=0.

3 M

1 (b) Find the series solution of (1+x²)y''+xy'-9y=0.

7 M

2 (a) (i) Solve the differential equation using the method variation of parameter y'+9y=sec3x.

4 M

2 (a) (ii) Solve the differential equation (D²-2D+1)y=10e^x.

3 M

Answer any one question from Q2 (b) & Q2 (c)

2 (b) Using the method of separation of variables, solve

\frac{\partial u}{\partial x} = 2 \frac{\partial u}{\partial t} + u; u (x, 0) = 6 e^{- 3 x} .

$\dfrac {\partial u}{\partial x}= 2 \dfrac {\partial u}{\partial t}+u; \ u(x,0)= 6e^{-3x}.$

7 M

2 (c) Find the series solution of 2x(x-1)y''-(x+1)y'+y=0; x₀=0.

7 M

Answer any two question from Q3 (a), (b) & Q3 (c), (d)

3 (a) Find the Fourier series for \[ f(x)= \left\{\begin{matrix}\pi + x; &-\pi

7 M

3 (b) (i) Find the Half range Cosine Series for f(x)=(x-1)²; 0

4 M

3 (b) (ii) Find the Fourier sine series for f(x)=e^x; 0

3 M

3 (c) Find the Fourier series for \[ f(x)= \left\{\begin{matrix}-\pi &-\pi

7 M

3 (d) (i) Find the Fourier series for f(x)=x²; 0

4 M

3 (d) (ii) Find the Fourier sine series for f(x)=2x; 0

3 M

Answer any two question from Q4 (a), (b) & Q4 (c), (b)

4 (a) (i) Prove that

i) \ L(e^{at})= \dfrac {1}{s-a}; s>a \\ ii) \ L(\sin h \ at) = \dfrac {a} {s^2-a^2).

$i) \ L(e^{at})= \dfrac {1}{s-a}; s>a \\ ii) \ L(\sin h \ at) = \dfrac {a} {s^2-a^2).$

4 M

4 (a) (ii) Find the Laplace transform of t sin 2t.

3 M

4 (b) (i) Using convolution theorem, Obtain the value of

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

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

4 M

4 (b) (ii) Find the inverse Laplace transform of

\frac{1}{(s - 2) (s + 3)} .

$\dfrac {1} {(s-2)(s+3)}.$

3 M

4 (c) Solve the initial value problem using Laplace transform:
y''+3y'+2y=e', y(0)=1, y'(0)=0.

7 M

4 (d) (i) Find the Laplace transform of \[ f(t)=f(t)= \left\{\begin{matrix}0; &0 \pi \end{matrix}\right. \]

4 M

4 (d) (ii) Evaluate t*e^t.

3 M

Answer any two question from Q5(a), (b) & Q5 (c), (d)

5 (a) Using Fourier integral representation prove that

\int_{0}^{\infty} \frac{\cos λ x + λ \sin λ x}{1 + λ^{2}} d y = {\begin{matrix} 0 & i f & x < 0 \\ \frac{π}{2} & i f & x = 0 \\ π e^{- x} & i f & x > 0 \end{matrix}

$\int^{\infty}_0 \dfrac {\cos \lambda x + \lambda \sin \lambda x}{1+\lambda ^2} dy = \left\{\begin{matrix} 0 &if &x<0 \\ \frac {\pi}{2} & if &x=0 \\ \pi e^{-x} & if &x>0 \end{matrix}\right.$

7 M

5 (b) (i) Form the partial differential equation by eliminating the arbitrary functions from f(x+y+z, x²+y²+z²)=0.

4 M

5 (b) (ii) Solve the following partial differential equation (z-y)p+(x-z)q=y-x.

3 M

5 (c) A homogeneous rod of conducting material of length 100 cm has its ends kept at zero temperature and the temperature initially is

u (x, 0) = {\begin{matrix} x; & 0 \leq x \leq 50 \\ 100 - x; & 50 \leq x \leq 100 \end{matrix}

$u(x,0)=\left\{\begin{matrix} \ \ \ \ x \ \ \ \ ; & 0\le x \le 50 \\ 100-x; & 50\le x \le 100 \end{matrix}\right.$

7 M

5 (d) (i) Solve \[ \dfrac {\partial^2z} {\partial x^2} + 3\dfrac {\partial^2z}{\partial x \partial y} + 2\dfrac {\partial ^2z}{\partial y^2} = x+y.

4 M

5 (d) (ii) Solve p-x²=q+x².

3 M

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