## 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}=x^{3}+y^{3}.(3 marks)
**2 (b) (i)** Find the Laplace transforms of function f(t)=t^{5}+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) cos^{3} t (ii) sin^{2} 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 P_{0}(x),P_{1}(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-x^{2} 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}>a.
\end{matrix}$$</a;><>(7 marks)