Advanced Engineering Mathematics - Jun 2014

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) Find the general solution of the differential equation y'=e^2x+3y(2 marks) 1 (a) (ii) Find the particular solution of the differential equation y?+4y=2sin3x by using method of undetermined coefficients.(2 marks) 1 (a) (iii) Find the inverse Laplace transform of following function: $$\dfrac{s}{s^{2}-3s+2}$$(3 marks) 1 (b) (i) i) Define Ordinary Point of the differential equation y''+P(x)y'+Q(x)y=0(2 marks) 1 (b) (ii) Find the value of $$\Gamma \left ( \dfrac34\Big|\dfrac14 \right )$$(2 marks) 1 (b) (iii) Express the function f(x)= x as a Fourier series in interval[-π,π](3 marks) 2 (a) (i) i) Evaluate $$\int_{0}^{\infty }\limits x^{2}e^{-x^{4}}dx$$(2 marks) 2 (a) (ii) Solve: (D^4>-1)y=0.(2 marks) 2 (a) (iii) Solve the partial differential equation u_xy=x³+y³.(3 marks) 2 (b) (i) Find the Laplace transforms of function f(t)=t⁵+cos5t+e^-100t.(3 marks) 2 (b) (ii) Using method of variation of parameters solve the differential equation
y?+4y=tan2x.(4 marks) 3 (a) Find the Laplace transforms of following functions: (i) cos³ t (ii) sin² t .(7 marks) 3 (b) State Convolution Theorem and using it find inverse Laplace transform of function $$f(t)=\dfrac{s^{2}}{(s^{2}+4)(s^{2}+9)}$$(7 marks) 3 (c) Using Laplace transform solve the differential equation:
y''+5y'+6y=e^-2,y(0)=0,y'(0)=-1.(7 marks) 3 (d) Evaluate i) $$\int_{-1}^{1}\limits \left ( 1-x^{2} \right )}^{n}dx$$ where n is a positive integer.
ii) $$\int_{0}^{\pi/2}\limits \sqrt{\sin \theta d \theta}\times \int_{0}^{\pi/2}\limits \dfrac{1}{\sqrt{\sin \theta}}d \theta$$(7 marks) 4 (a) i) Prove that $$J_{\dfrac{3}{2}}(x)=\sqrt{\dfrac{2}{\pi x}}\left [ \dfrac{\sin x}{x}- \cos x \right ]$$
ii) p_n(-1)ⁿ=(-1)ⁿ.(7 marks) 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 marks) 4 (c) i) Solve the differential equation $$xdy-ydx=\sqrt{x^{2}+y^{2}}$$
ii) Solve: $$x^{3}\dfrac{d^{3}y}{dx^{3}}+2x^{2}\dfrac{d^{2}y}{dx^{2}}+2y=10\left ( x+\dfrac{1}{x} \right )$$.(7 marks) 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 marks) 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 marks) 5 (b) i) Evaluate $$\int_{0}^{1}\limits (x \log x)^{3}dx$$.
ii) By using the relation between Beta and Gamma function prove that
$$\beta (m,n)\beta(m+n,p)\beta(m+n+p,q)=\dfrac{\Gamma m\Gamma n\Gamma p\Gamma q}{\Gamma (m+n+p+q)}$$(7 marks) 5 (c) Solve Completely the equation $$\dfrac{\partial^2 y }{\partial x^2}=c^{2}\dfrac{\partial^2 y }{\partial x^2}$$ representing the vibrations of a string of length l fixed at both ends given that,
$$y(0,t)=y(l,t)=0,y(x,0)=f(x),\dfrac{\partial y}{\partial t}(x,0)=0,0\ltx\ltl$$.< a="">

</x<l$$.\lt\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (d)\lt/b\gt Find the Fourier Transform of the function f defined as follows: \ltbr\gt $$f(x)=\begin{matrix} x &\left | x \right | &<a; \\0="" &\left="" |="" x="" \right="" &="">a. \end{matrix}$$</a;></span>(7 marks)