GTU Civil Engineering (Semester 3)
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 $\dfrac {dy}{dx}+\dfrac {1}{x} = \dfrac {e^y}{x^2}$
4 M
1 (a) (ii) Solve the differential equation yex dx+(2y+ex)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=10ex.
3 M
Answer any one question from Q2 (b) & Q2 (c)
2 (b) Using the method of separation of variables, solve $\dfrac {\partial u}{\partial x}= 2 \dfrac {\partial u}{\partial t}+u; \ u(x,0)= 6e^{-3x}.$
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)=ex; 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).$
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 $L^{-1}\left \{ \dfrac {1}{s(s^2+4)} \right \}$
4 M
4 (b) (ii) Find the inverse Laplace transform of $\dfrac {1} {(s-2)(s+3)}.$
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*et.
3 M

Answer any two question from Q5(a), (b) & Q5 (c), (d)
5 (a) Using Fourier integral representation prove that $\int^{\infty}_0 \dfrac {\cos \lambda x + \lambda \sin \lambda x}{1+\lambda ^2} dy = \left\{\begin{matrix} 0 &if &x<0 \\ \frac {\pi}{2} & if &x=0 \\ \pi e^{-x} & if &x>0 \end{matrix}\right.$
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 $u(x,0)=\left\{\begin{matrix} \ \ \ \ x \ \ \ \ ; & 0\le x \le 50 \\ 100-x; & 50\le x \le 100 \end{matrix}\right.$
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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