Advanced Engineering Mathematics - Jun 2015

Computer Engineering (Semester 3)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks. 1 (a) (i) Solve the differential equation $$ \dfrac {dy}{dx}+\dfrac {1}{x} = \dfrac {e^y}{x^2} $$(4 marks) 1 (a) (ii) Solve the differential equation ye^x dx+(2y+e^x)dy=0.(3 marks) 1 (b) Find the series solution of (1+x²)y''+xy'-9y=0.(7 marks) 2 (a) (i) Solve the differential equation using the method variation of parameter y'+9y=sec3x.(4 marks) 2 (a) (ii) Solve the differential equation (D²-2D+1)y=10e^x.(3 marks)

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 marks) 2 (c) Find the series solution of 2x(x-1)y''-(x+1)y'+y=0; x₀=0.(7 marks)

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

3 (a) Find the Fourier series for $$ f(x)= \left\{\begin{matrix}\pi + x; &-\pi \ltx\lt0 \\\pi="" -="" x;="" &="" 0="" \lt="" x\lt="" \pi="" \end{matrix}\right.$$<="" a="">

</x<0>(7 marks) 3 (b) (i) Find the Half range Cosine Series for f(x)=(x-1)²; 0<x<1.< a="">

</x<1.<></span>(4 marks) 3 (b) (ii) Find the Fourier sine series for f(x)=e^x; 0<x<π< a="">

</x<π<></span>(3 marks) 3 (c) Find the Fourier series for $$ f(x)= \left\{\begin{matrix}-\pi &-\pi \ltx\lt0 \\="" x-\pi;="" &="" 0="" \lt="" x\lt="" \pi="" \end{matrix}\right.$$<="" a="">

</x<0>(7 marks) 3 (d) (i) Find the Fourier series for f(x)=x²; 0<x<π< a="">

</x<π<></span>(4 marks) 3 (d) (ii) Find the Fourier sine series for f(x)=2x; 0<x<1.< a="">

</x<1.<></span>(3 marks)

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 marks) 4 (a) (ii) Find the Laplace transform of t sin 2t.(3 marks) 4 (b) (i) Using convolution theorem, Obtain the value of $$ L^{-1}\left \{ \dfrac {1}{s(s^2+4)} \right \} $$(4 marks) 4 (b) (ii) Find the inverse Laplace transform of $$ \dfrac {1} {(s-2)(s+3)}.$$(3 marks) 4 (c) Solve the initial value problem using Laplace transform:
y''+3y'+2y=e', y(0)=1, y'(0)=0.(7 marks) 4 (d) (i) Find the Laplace transform of $$ f(t)=f(t)= \left\{\begin{matrix}0; &0\ltt\lt\pi \\\sin="" t="" &t=""\gt \pi \end{matrix}\right. $$</t<\pi>(4 marks) 4 (d) (ii) Evaluate t*e^t.(3 marks)

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 marks) 5 (b) (i) Form the partial differential equation by eliminating the arbitrary functions from f(x+y+z, x²+y²+z²)=0.(4 marks) 5 (b) (ii) Solve the following partial differential equation (z-y)p+(x-z)q=y-x.(3 marks) 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 marks) 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 marks) 5 (d) (ii) Solve p-x²=q+x².(3 marks)