STUPIDSID
1 (a)
Solve y''+4y'-12y=e2x-3sin 2x.
6 M
1 (b)
By the method undetermined coefficients solve \( \dfrac {d^2y}{dx^2}+ y = 2 \cos x. \)
7 M
1 (c)
Solve by the method of variation of parameter y''+4y=tan 2x.
7 M
2 (a)
Solve \( \dfrac {d^6y}{dx^4}+ m^4 y=0. \)
6 M
2 (b)
Solve (D2+7D+12)y=cos hx.
7 M
2 (c)
By the method variation of parameters, solve y''+y=x sin x.
7 M
3 (a)
Solve the simultaneous equations \[ \dfrac {dx}{dt}+ 2y + sin t=0, \ \ \ \dfrac {dy}{dt}-2x - \cos t=0 \] given that x=0 and y=1 when t=0.
7 M
3 (b)
Solve x2 y''-xy'+2y=x sin (log x).
7 M
3 (c)
Solve \( \dfrac {dy}{dx} - \dfrac {dx}{dy} = \dfrac {x}{y} - \dfrac {y}{x} . \)
6 M
4 (a)
Solve (x+a)2 y''-4(x+a)y'+6y=x.
7 M
4 (b)
Solve \( P=\tan \left ( x - \dfrac {p}{1+p^2} \right ) . \)
7 M
4 (c)
Find the general and the singular solution of the equation y=px+p3.
6 M
5 (a)
Form the Partial Differential Equation of z=y f(x)+x g(y), where f and g are arbitrary functions.
7 M
5 (b)
Derive one dimensional heat equation.
7 M
5 (c)
Evaluate \( \displaystyle \int^\infty_{0}\int^\infty_0 e^{-(x^2 + y^2)} dx \ dy \) by changing into polar co-ordinates.
6 M
6 (a)
Solve \( \dfrac {\partial^1 Z}{\partial x \partial y}= \sin x \sin y, \text { for which } \dfrac {\partial z}{\partial y}=-2 \sin y \) when x=0 and z=0, when y is an odd multiple of π/2.
7 M
6 (b)
Evaluate \( \displaystyle \iint_R \ xydxdy, \) where R is the region bounded by x-axis, the ordinate x=2a and the parabola x2=4 ay.
7 M
6 (c)
Evaluate \( \displaystyle \int^c_{-c} \int^b_{-b} \int^a_{-a} (x^2+y^2 + z^2) \ dz \ dy \ dx. \)
6 M
7 (a)
Define Gamma function Beta function. Prove that \( \Gamma(1/2) \sqrt{\pi} \)
7 M
7 (b)
Express the vector \( \overline{F}=z\widehat{i}-2x\widehat{j}+y\widehat{k} \) in cylindrical co-ordinates.
6 M
7 (c)
Find the volume common on the cylinders x2+y2=a2 and x2+z2=a2.
7 M
8 (a)
Prove that \( \beta (m, n) = \dfrac {\Gamma m \Gamma n}{\Gamma{(m+n)}} \)
7 M
8 (b)
Show that the area between the parabolas y2=4ax and x2=4ay is \( \dfrac {16}{3} \)a2.
6 M
8 (c)
Prove that the cylindrical co-ordinate system is orthogonal.
7 M
9 (a)
Find L{e-2t sin 3t+et t cost}.
7 M
9 (b)
Find the inverse Laplace transform of \( \dfrac {4s+5}{(s-1)^2(s+2)} \).
6 M
9 (c)
Solve y''-6y' + 9y=12t2 e-3t by Laplace transform method with y(0)=0=y(0).
7 M
10 (a)
Express \( f(t) = \begin{bmatrix}
\cos t &0 2\pi \end{bmatrix} \) in terms of unit step function and hence find its Laplace transform.
7 M
10 (b)
Solve by Laplace transform y''+6y'+9y=12t2 e-3t with y(0)=0=y'(0).
6 M
10 (c)
Find \( L \left \{ \dfrac {\cos at - \cos bt}{t} \right \} \).
7 M
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