STUPIDSID
Answer the following one mark each questions
1(a)
Integreating factor of the differential equation
\( \dfrac{dx}{dy}+\dfrac{3x}{y}=\dfrac{1}{y^2} \) is _______
\( \dfrac{dx}{dy}+\dfrac{3x}{y}=\dfrac{1}{y^2} \) is _______
1 M
1(b)
The general solution of the differential equation \( \dfrac{dy}{dx}+\dfrac{y}{x}=\tan 2x \) _______.
1 M
1(c)
The orthogonal trajectory of the family of curve x2 + y2 = c2 is _______ .
1 M
1(d)
Particular integral of (D2 + 4)y = cos 2x is _______ .
1 M
1(e)
X=0 is a regular singular point of
\( 2x^2y''+3xy'(x^2-4)y=0\ \text{say true or false} \)
\( 2x^2y''+3xy'(x^2-4)y=0\ \text{say true or false} \)
1 M
1(f)
The solution of
(y ' z)p + (z ' x)q = x ' y is _______
(y ' z)p + (z ' x)q = x ' y is _______
1 M
1(g)
State the type ,order and degree of differential equation \( \left ( \dfrac{dx}{dy} \right )^2+5y^{\dfrac{1}{3}}=x \) is _______
1 M
1(h)
Solve (D+D')z=cos x
1 M
1(i)
Is the partial differential equation \[2\dfrac{\partial ^2 u}{\partial x^2}+4\dfrac{\partial ^2 u}{\partial x \partial y}+3\dfrac{\partial ^2 u}{\partial y^2}=6\ \text {elliptic}?\]
1 M
1(j)
\( L^{-1}\left ( \dfrac{1}{(s+a)^2} \right )= \) _______
1 M
1(k)
If f(t) is a periodic function with period t L [f(t)] = _______ .
1 M
1(l)
Laplace transform of f(t) is defined for +ve and 've values of t. Say true or false.
1 M
1(m)
State Duplication (Legendre) formula.
1 M
1(n)
Find \( B\left ( \dfrac{9}{2},\dfrac{7}{2} \right ) \)
1 M
2(a)
Solve : 9y y' + 4x = 0
3 M
2(b)
Solve : \(\dfrac{dy}{dx}+y \cot x=2 \cos x\)
4 M
Solve any one question from Q.2(c) & Q.2(d)
2(c)
Find series solution of y'' + xy = 0
7 M
2(d)
Determine the value of \( (a)J\frac{1}{2}(x)\ \ \ \ (b)J\frac{3}{2}(x) \)
7 M
Solve any three question from Q.3(a), Q.3(b), Q.3(c) & Q.3(d), Q.3(e), Q.3(f)
3(a)
Solve (D2 + 9)y = 2sin 3x + cos 3x
3 M
3(b)
Solve y'' + 4y' = 8x2 by the method of undetermined coefficients.
4 M
3(c)
(i) Solve x2p + y2q = z2
(ii) Solve by charpit's method px+qy = pq
(ii) Solve by charpit's method px+qy = pq
7 M
3(d)
Solve y'' + 4y' + 4 = 0 , y(0) = 1 , y'(0) = 1
3 M
3(e)
Find the solution of y'' + a2y' = tan ax , by the method of variation of parameters.
4 M
3(f)
Solve the equation ux = 2ut + u given u(x,0)=4e-4x by the method of seperation of variable.
7 M
Solve any three question from Q.4(a), Q.4(b), Q.4(c) & Q.4(d), Q.4(e), Q.4(f)
4(a)
Find the fourier transform of the function f(x) = e-ax2
3 M
4(b)
Obtain fourier series to represent f(x) =x2 in the interval
\( -\pi
\( -\pi
4 M
4(c)
Find Half-Range cosine series for \[F(x)=\begin{matrix}
kx & ,0\leq x\leq \dfrac{1}{2}\\
k(l-x) & ,\dfrac{l}{2}\leq x\leq l
\end{matrix}\]
Also prove that \( \sum {^\infty_{n=1}}=\dfrac{1}{(2n-1)^2}=\dfrac{\pi^2}{8} \)
Also prove that \( \sum {^\infty_{n=1}}=\dfrac{1}{(2n-1)^2}=\dfrac{\pi^2}{8} \)
7 M
4(d)
Expres the function \[F(x)=\begin{matrix}
2 & ,|x|<2\\
0 & ,|x|>2\\
\end{matrix}\ \ \ \text{as Fourier integral}.\]
3 M
4(e)
Find the fourier series expansion of the function \[F(x)=\begin{matrix}
-\pi & -\pi
4 M
4(f)
Find fourier series to represent the function
F(x) = 2x-x2 in 0 < x < 3
F(x) = 2x-x2 in 0 < x < 3
7 M
Solve any three question from Q.5(a), Q.5(b), Q.5(c) & Q.5(d), Q.5(e), Q.5(f)
5(a)
Find \( L{-1}\left \{ \dfrac{1}{(s+\sqrt{2})(s-\sqrt{3})} \right \} \)
3 M
5(b)
Find the laplace transform of \[(i)\dfrac{\cos at-\cos bt}{t}\] \[(ii)t\sin at\]
4 M
5(c)
State convolution theorem and use to it evaluate \[L^{-1}\left \{ \dfrac{1}{(s^2+a^2)^2} \right \}\]
7 M
5(d)
\( (a)L\left \{ t^2 \cos h 3t\right \} \)
3 M
5(e)
Find \( L^{-1}\left \{ \dfrac{1}{s^4-81} \right \} \)
4 M
5(f)
Solve the equation y'' ' 3y' + 2y = 4t + e3t , when y(0)=1 , y'(0) = -1
7 M
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