1(a)
Expand π-x

^{2}in a half range since series in the interval (0,π) up to the first terms.
2 M

1(b)
Find the Fourier transform of Dirac Delta Function δ(t-a).

2 M

1(c)
Find the function whose sine transform is \(\dfrac{e^{-s}}{s}\)

3 M

Solve any one question from Q.1(d) & Q.1(e)

1(d)
Find the Fourier series to represent the function f(x) given by

\[f(x)=\left\{\begin{matrix} 0\ \ \ for\ \ -\pi\leq x\leq 0\\ \sin x \ \ \ for \ \ 0 \leq x \leq \pi \end{matrix}\right.\]

Deduce that \[\dfrac{1}{1.3}-\dfrac{1}{3.5}+\dfrac{1}{5.7}-\cdots\cdots=\dfrac{\pi-2 }{4}\]

\[f(x)=\left\{\begin{matrix} 0\ \ \ for\ \ -\pi\leq x\leq 0\\ \sin x \ \ \ for \ \ 0 \leq x \leq \pi \end{matrix}\right.\]

Deduce that \[\dfrac{1}{1.3}-\dfrac{1}{3.5}+\dfrac{1}{5.7}-\cdots\cdots=\dfrac{\pi-2 }{4}\]

7 M

1(e)
Develop \(\displaystyle \sin\left ( \dfrac{\pi x}{l} \right )\) in a half-range cosine series in the range 0<x<1.

7 M

2(a)

Find L{F(t)} if \(\displaystyle F(t)=\left\{\begin{matrix} \sin \left ( t-\dfrac{\pi }{3} \right ),& t> \dfrac{\pi }{3} \\\\ 0, & t<\dfrac{\pi }{3}\\ \end{matrix}\right.\)

2 M

2(b)
Define Unit Step Function and Find its Laplace Transform.

2 M

2(c)
Prove that :\(L^{-1}\left ( \dfrac{s}{s^{4}+s^{2}+1} \right )=\dfrac{2}{\sqrt{3}}sinh\dfrac{t}{2}sin\dfrac{\sqrt{3}}{2}t\)

3 M

Solve any one question from Q.2(d) & Q.2(e)

2(d)
State convolution theorem and hence evaluate

\[L^{-1}\left \{ \dfrac{p}{(p^2+1)(p^2+4)} \right \}.\]

\[L^{-1}\left \{ \dfrac{p}{(p^2+1)(p^2+4)} \right \}.\]

7 M

2(e)
Solve the simultaneous equations using Laplace Transform:

\(\dfrac{dx}{dt}-y=e^t,\dfrac{dy}{dt}+x=sint,given x(0)=1,y(0)=0\)

\(\dfrac{dx}{dt}-y=e^t,\dfrac{dy}{dt}+x=sint,given x(0)=1,y(0)=0\)

7 M

3(a)
Solve the differential equation by Removal of first derivative method.

\[\dfrac{d^{2}y}{dx^{2}}-4x\dfrac{dy}{dx}+(4x^2-1)y=-3e^{x^2}\sin2x\]

\[\dfrac{d^{2}y}{dx^{2}}-4x\dfrac{dy}{dx}+(4x^2-1)y=-3e^{x^2}\sin2x\]

2 M

3(b)
Solve by changing the independent variable

\[(1+x^2)^2\dfrac{d^{2}y}{dx^{2}}+2x(1+x^2)\dfrac{dy}{dx}+4y=0\]

\[(1+x^2)^2\dfrac{d^{2}y}{dx^{2}}+2x(1+x^2)\dfrac{dy}{dx}+4y=0\]

2 M

3(c)
Solve by the method of variation of parameters:

\[\dfrac{d^{2}y}{dx^{2}}-y=\dfrac{2}{1+e^x}\]

\[\dfrac{d^{2}y}{dx^{2}}-y=\dfrac{2}{1+e^x}\]

3 M

Solve any one question from Q.3(d) & Q.3(e)

3(d)
Solve in series the equation

\[2x(1-x)\frac{d^2y}{dx^2}+(5-7x)\frac{dy}{dx}-3y=0\]

\[2x(1-x)\frac{d^2y}{dx^2}+(5-7x)\frac{dy}{dx}-3y=0\]

7 M

3(e)
Solve in series the equation

\[(1-x^2)\dfrac{d^2y}{dx^2}-2x\dfrac{dy}{dx}+2y=0\]

\[(1-x^2)\dfrac{d^2y}{dx^2}-2x\dfrac{dy}{dx}+2y=0\]

7 M

4(a)
Solve the differertial equation

\[(z^2-2yz-y^2)p+(xy+zx)q=xy-zx\]

\[(z^2-2yz-y^2)p+(xy+zx)q=xy-zx\]

2 M

4(b)
Solve \[p^2-q^2=x-y\]

2 M

4(c)
Solve \[(D^2+5DD'+6D'^2)z=\dfrac{1}{y-2^x}\]

3 M

Solve any one question from Q.4(d) & Q.4(e)

4(d)
If a string of length l is initially at rest in equilibrium position and each of its points is given the velocity

\[\left ( \dfrac{dy}{dt} \right )_{t=0}=bsin^{3}\dfrac{\pi x}{l}\],find the displacement y(z,t).

\[\left ( \dfrac{dy}{dt} \right )_{t=0}=bsin^{3}\dfrac{\pi x}{l}\],find the displacement y(z,t).

7 M

4(e)
A bar with insulated sides is initially at a temperature 0°C,throughout. The end x=0 is kept at 0°C,and heat is suddenly applied at the end x=l so that \(\displaystyle \dfrac{\partial u}{\partial x}=A\) for x=1, where A is a constant. Find the temperature u(x,t).

7 M

5(a)

Find the directional derivative of\(\displaystyle \phi =5x^2y-5y^2z+\dfrac{5}{2}z^2x\)at the point P(1,1,1) in the direction of the line\(\displaystyle \dfrac{x-1}{2}=\dfrac{y-3}{-2}=\dfrac{z}{1}\)

2 M

5(b)
Prove that vector \[f(r)\vec{r}\] is irrotational

2 M

5(c)

A Vector field is given by\[ \overline{F}=(\sin y)\hat{i}+x(1+\cos y)\hat{j}\].Evaluate the line integral over the circular path given by

\[x^2+y^2=a^2,z=0.\]

3 M

Solve any one question from Q.5(d) & Q.5(e)

5(d)
Verify Stoke's theorem for the vector \(\overline{F}=z\hat{i}+x\hat{j}+y\hat{k}\)taken over the half of the sphere \(x^{2}+y^{2}=a^{2},z=0\) lying above the xy-plane.

7 M

5(e)
Evaluate\[ \iint_s \overline {A}\cdot\hat{n}\ ds\ \text{where}\ \overline{A}=z\hat{i}+x\hat{j}-3y^{2}z\hat{k}\]and S is the surface of the cylinder x

^{2}+y^{2}=16 included in the frst octant between z=0 and z=5.
7 M

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