MORE IN Engineering Maths 2
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INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) Solve $4\dfrac{d^{4}y}{dx^{4}}-4\dfrac{d^{3}y}{dx^{3}}-23\dfrac{d^{2}y}{dx^{2}}+12\dfrac{dy}{dx}+36y=0$
6 M
1 (b) Solve $\dfrac{d^{3}y}{dx^{3}}+6\dfrac{d^{2}y}{dx^{2}}+11\dfrac{dy}{dx}+6y=e^{x}+1$ using inverse differential operator method
7 M
1 (c) Solve (D2-2D)y=ex sinx using method of undetermined coefficients
7 M

2 (a) Solve (4D4-8D3-7D2+11D+6)y=0
6 M
2 (b) Solve (D2+4)y=x2+ex using inverse differential operator method
7 M
2 (c) Solve (D2-2D+2)y=ex tan x using method of variation of parameters
7 M

3 (a) Solve $\dfrac{dx}{dt}-7x+y=0,\dfrac{dy}{dt}-2x-5y=0$
6 M
3 (b) Solve $x^{2}\dfrac{d^{2}y}{dx^{2}}+4x\dfrac{dy}{dx}+2y=e^{x}$
7 M
3 (c) Solve y=2px+y2p3 by solving for x
7 M

4 (a) Solve $(1+x)^{2}\dfrac{d^{2}y}{dx^{2}}+(1+x)\dfrac{dy}{dx}+y=2$sin (log(1+x))\]
6 M
4 (b) Solve $\dfrac{\mathrm dy}{\mathrm d x}-\dfrac{\mathrm dx}{\mathrm d y}=\dfrac{x}{y}-\dfrac{y}{x}$ by solving for P.
6 M
4 (c) Solve (px-y)(py+x)=a2p by reducing to Clairaut's form.
7 M

5 (a) From the function f(x2+y2,z-xy)=0 from the partial differential equation.
6 M
5 (b) Derive one dimensional wave equation as $\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}$
7 M
5 (c) Evaluate $\int_{0}^{1}\limits\int_{x^2}^{2-x}\limits xy\ \mathrm d y \ \mathrm d x$ by changing the order of integration
7 M

6 (a) Solve $\frac{\partial^2u}{\partial x\partial y}=\sin x\ \sin y for \ which\ \frac{\partial u }{\partial y}=-2 \sin y$when x=0 and u=0 when y is an odd multiple of $\dfrac{\pi }{2}$
6 M
6 (b) Derive one dimensional heat equation as $\dfrac{\partial u}{\partial t}=c^{2}\dfrac{\partial^2u }{\partial x^2}$
7 M
6 (c) Evaluate $\int_{-1}^{1}\limits\int_{y}^{y}\limits\int_{x+y}^{x-y}\limits\ (x+y+z)\ dydxdz$
7 M

7 (a) Find the area between the parabolas y2=4ax and x2=4ay using double integral
6 M
7 (b) Evaluate$\int_{0}^{1} \limits\dfrac{dx}{\sqrt{1-x^{4}}}$ using beta and gamma functions
7 M
7 (c) Express the vector zi-2xj+yk in cylindrical coordinates
7 M

8 (a) Find the volume of the solid bounded by the planes x=0, y=0, x+y+z=1 and z=0 using triple integral
6 M
8 (b) Express $\int_{0}^{\pi/2}\limits \sqrt{\sin \theta}\ d\theta \times \int_{0}^{\pi/2}\limits\dfrac{d\theta}{\sqrt{\sin \theta}}$ using beta and gamma functions
7 M
8 (c) Express the vector field 2yi-zj +3xk in spherical polar coordinate system
7 M

9 (a) Find Laplace transform of $te^{-4t}\sin3t \ and \ \dfrac{e^u-e^{-u}}{t}$
6 M
9 (b) Using f(t) in terms of unit step function and find its Laplace transform given that
$\left\{\begin{matrix} t^2, &04 \end{matrix}\right.$
7 M
9 (c) Find $L^{-1}\left \{ \dfrac{1}{(s+1)(s^2+9)} \right \}$ using convolution theorem
7 M

10 (a) A periodic function f(t) with period 2 is defined by $f(t)=\left\{\begin{matrix} t, &0 6 M 10 (b) Find \[L^{-1}\left \{ \dfrac{5s-2}{3s^2+4s+8}+log\left ( \dfrac{1} {s^2}-1\right ) \right \}$
7 M
10 (c) Solve using Laplace transform method $\dfrac{d^2y}{dt^2}+2\dfrac{dy}{dt}+y=te^{-1}\ with \ y(0)=1,y^1(0)=2$
7 M

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