STUPIDSID
Answer the following one mark each questions:
1(a)
Find \[\Gamma \left ( \dfrac{13}{2} \right )\]
1 M
1(b)
State relationship between beta and gamma functions.
1 M
1(c)
Represent graphically the given saw-tooth function f(x) = 2x, 0 ≤
x < 2 and f(x + 2) = f(x) for all x.
1 M
1(d)
For a periodic function f with fundamental period p, state the formula to
find Laplace transform of f.
1 M
1(e)
Find \[L\left ( e^{-3t}f(t) \right )\], if \[L\left ( f(t) \right )=\dfrac{s}{(s-3)^2}\].
1 M
1(f)
Find \[L[(2t-1)^2]\].
1 M
1(g)
Find the extension of the function f(x) = x + 1, define over (0,1] to
[?1, 1] - {0} which is an odd function.
1 M
1(h)
Is the function \[f(x)=\left\{\begin{matrix}
x & 0\leq x\leq 2\\
x^2 & 2
1 M
1(i)
Is the differential equation \[\dfrac{dy}{dx}=\dfrac{y}{x}\] exact? Give reason.
1 M
1(j)
Give the differential equation of the orthogonal trajectory to the equation y = cx2 .
1 M
1(k)
If y = c 1 y1+c2 y 2 = ex (c1 cos x + c 2 sin x) is a complementary function of a second order differential equation, find the Wronskian W(y1 , y2 ).
1 M
1(l)
Solve (D2 + D + 1)y = 0; where \[D=\frac{\mathrm{d} }{\mathrm{d} t}\].
1 M
1(m)
Is \[u(t,x)=50e^{(t-x)/2}\], a solution to \[\dfrac{\partial u}{\partial t}=\dfrac{\partial u}{\partial x}+u\] ?
1 M
1(n)
Give an example of a first order partial differential equation of Clairaut's
Form.
1 M
2(a)
Solve: \[\dfrac{dy}{dx}=\dfrac{x^2-x-y^2}{2xy}\].
3 M
2(b)
Solve \[\dfrac{dy}{dx}+\dfrac{1}{x}y=x^3y^3\].
4 M
Solved any one question from Q.2(c) & Q.2(d)
2(c)
Find the series solution of \[(x-2)\dfrac{d^2y}{dx^2}-x^2\dfrac{dy}{dx}+9y=0\] about x0 = 0.
7 M
2(d)
Explain regular-singular point of a second order differential equation and
find the roots of the indicial equation to x2y''+xy'-(2-x)y=0.
7 M
Solved any one question from Q.3 & Q.4
3(a)
Find the complete solution of \[\dfrac{d^3y}{dx^3}+8y=\cos h(2x)\].
3 M
3(b)
Find solution of \[\dfrac{d^2y}{dx^2}+9y=\tan 3x\], using the method of variation of parameters.
4 M
3(c)
Using separable variable technique find the acceptable general solution to the one-dimensional heat equation \[\dfrac{\partial u}{\partial t}=c^2\dfrac{\partial ^2 u}{\partial x^2}\] and find the solution satisfying the conditions u(0, t) = u(π , t) = 0 for t > 0 and u(x, 0) =π- x, 0 < x < π ..
7 M
4(a)
Solve completely, the differential equation
\[\dfrac{d^2y}{dx^2}-6\dfrac{dy}{dx}+9=\cos(2x)\sin x\] .
\[\dfrac{d^2y}{dx^2}-6\dfrac{dy}{dx}+9=\cos(2x)\sin x\] .
3 M
4(b)
Solve completely the differential equation \[x^2\dfrac{d^2y}{dx^2}-6x\dfrac{dy}{dx}+6y=x^{-3}\log x\] .
4 M
4(c)i
Form the partial differential equation for the equation (x-a)(y-b)-z2=x2+y2.
3 M
4(c)ii
Find the general solution to the partial differential equation xp+yp=x-y.
3 M
Solved any one question from Q.5 & Q.6
5(a)
Find the Fourier cosine integral of \[f(x)=\dfrac{\pi}{2}e^{-x},x\geq 0\].
3 M
5(b)
For the function \[f(x)=\cos 2x\], find its Fourier sine series over [0, π].
4 M
5(c)
For the function \[f(x)=\left\{\begin{matrix}
x & 0\leq x\leq 2\\
4-x; & 2\leq x\leq 4
\end{matrix}\right.\], find its Fourier series. Hence show that \[\dfrac{1}{1^2}+\dfrac{1}{3^2}+\dfrac{1}{5^2}+\cdots=\dfrac{\pi^2}{16}\] .
7 M
6(a)
Find the Fourier cosine series of f(x)=e-x, Where 0≤ x≤ π .
3 M
6(b)
Find the Fourier cosine series of \[\int ^\infty _0 \dfrac{\lambda^3 \sin \lambda x}{\lambda^4+4}d\lambda=\dfrac{\pi}{2}e^{-x}\cos x,x>0\] .
4 M
6(c)
Is the function f(x) = x+|x|, -π ≤ x ≤ π even or odd? Find its Fourier
series over the interval mentioned.
7 M
Solved any one question from Q.7 & Q.8
7(a)
Find \[L\left \{ \int ^t_0e^u (u+\sin u)du \right \}\] .
3 M
7(b)
Find \[L^{-1}\left \{ \dfrac{1}{s(s^2-3s+3)} \right \}\].
4 M
7(c)
Solve the initial value problem: \[y''-2y'=e^t\sin t, y(0)=y'(0)=0\] using Laplace transform.
7 M
8(a)
Find \[L\left \{ t(\sin t-t \cos t) \right \}\].
3 M
8(b)
Find \[L^{-1}\left \{ \dfrac{e^{-2s}}{(s^2+2)(s^2-3)} \right \}\].
4 M
8(c)
State the convolution theorem and verify it for f(t) = t and g(t) = e2t.
7 M
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