STUPIDSID
1(10)
Give the differential equation of the orthogonal trajectory of the family of circles \[x^2+y^2=a^2.\]
1 M
1(11)
Find the Wronskian of the two function sin2x and cos2x.
1 M
1(12)
Solve \[\left ( D^2+6D+9 \right )x=0; D =\frac{d}{dt}.\]
1 M
1(13)
To solve heat equation \( \frac{\partial u}{\partial t}=C^2\frac{\partial^2u }{\partial x^2} \)/ how many initial and boundary conditions are required.
1 M
1(14)
From the partial differential equations from z = (x+at) +g(x-at).
1 M
1(2)
State relation between beta and gamma function.
1 M
1(3)
Define Heaviside's unit step function.
1 M
1(4)
Define Laplace transform of f(t),t≥0.
1 M
1(5)
Find Laplace transform of \[t^{\frac{-1}{2}}.\]
1 M
1(6)
Find \( L\left \{ \frac{Sinat}{t} \right \},\ \ \text{given that L}\left \{ \frac{Sint}{t} \right \} = \tan ^{-1}\left \{ \frac{1}{s} \right \}. \)/
1 M
1(7)
Find the continuous extension of the function \[f\left ( x \right )=\frac{x^2+x-2}{x^2-1}\ \text{to}\ \ x=1\]
1 M
1(8)
Is the function \( f\left ( x \right )=\frac{1}{x} \)/ continous on [-1, 1]? Give reason.
1 M
1(9)
Solve \[\frac{dy}{dx}=e^{3x-2y}+x^2e^{-2y}.\]
1 M
1(a)
Find ⌈\[\left ( \frac{1}{2} \right ).\]
1 M
2(a)
Solve:\[\left ( x+1 \right )\frac{dy}{dx}-y = e^{3x}(x+1).\]
3 M
2(b)
Solve: \[\frac{dy}{dx}+\frac{y\cos x+\sin y+y}{\sin x+x\cos y+x}=0\]
4 M
Solve any one question.Q2(c) &Q2(d)
2(c)
Fin the series solution of \[\frac{d^2y}{dx^2}+xy=0.\]
7 M
2(d)
Find the general solution of \(2x^2y''+xy'+\left ( x^2-1 \right )y=0 \)/ by using frobenius method.
7 M
solve any one question Q.3(a,b,c) &Q4(a,b,c)
3(a)
Solve : \[\left ( D^3-3D^2+9D-27 \right )y=\cos 3x.\]
3 M
3(b)
Solve : \[x^2\frac{d^2y}{dx^2}+4x\frac{dy}{dx}+2y=x^2\sin \left ( lnx \right ).\]
4 M
3(c)(i)
Solve : \[\frac{\partial ^3z}{\partial x^3}-2\frac{\partial^3z }{\partial x^2 \partial{y}}=2e^{2x}.\]
3 M
3(c)(ii)
Find the general solutiion to the partial differential equation \[\left ( x^2-y^2-z^2 \right )+2xyq=2xz.\]
4 M
4(a)
Solve : \[\left ( D^3-D \right )=x^3.\]
3 M
4(b)
Find the solution \( y''-3y'+2y=e^x, \)/ using the method of variation of parameters.
4 M
4(c)
Solve \( x\frac{\partial u}{\partial x}-2y\frac{\partial u}{\partial y}=0 \) / using method of separation of variables.
7 M
solve any one question Q.5(a,b,c) &Q6(a,b,c)
5(a)
Find the Fourier cosine integral of \[f\left ( x \right )= e^{kx},x>0, k>0\]
3 M
5(b)
Express \( f(x)=|x|,-\pi / as fouries series.
4 M
5(c)
Find Fourier Series for the function f(x) given by \( \left\{\begin{matrix}
1+\frac{2x}{\pi }; &-\pi \leq x\leq 0 \\
1-\frac{2x}{\pi } & 0\leq x\leq \pi
\end{matrix}\right. \)/ Hence deduce that \[\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+....\frac{\pi ^2}{8}.\]
7 M
6(a)
Obtain the Fourier Series of periodic function function \[f(x)=2x,-1
3 M
6(b)
Show that \[
4 M
6(c)
Expand f(x) in Fourier series in the interval (0,2π) if \( f(x)\left\{\begin{matrix}
-\pi ; &0,/ and hence show that \[\sum_{r=0 }^{\infty }\frac{1}{\left ( 2r+1 \right )^2}=\frac{\pi ^2}{8}.\]
7 M
solve any one question Q.7(a,b,c) &Q8(a,b,c)
7(a)
Find L {{\int ^t_0 e^t\frac{\sin t}{t}dt}}
3 M
7(b)
Find \[L^{-1}\left \{ \frac{2s^2-1}{\left ( s^2+1 \right )\left ( s^2
+4 \right )} \right \}\]
4 M
7(c)
Solve initial value problem : \( y''-3y'+2y=4t+e^{3t},y(0)=1 \)/ and y'(0)=-1, using Laplace transform.
7 M
8(a)
Find \[L\left \{ t\sin 3t\cos 2t \right \}.\]
3 M
8(b)
Find \[L^{-1}\left \{ \frac{e^{-35}}{s^2+8s+25} \right \}.\]
4 M
8(c)
State the convolution theorem and apply it to evaluate \[L^{-1}\left \{ \frac{s}{\left ( s^2+a^z \right )^z} \right \}.\]
7 M
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