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MU First Year Engineering (Semester 2)
Applied Mathematics 2
December 2016
Total marks: --
Total time: --
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 $\left [ \log \left ( x^2+y^2 \right )+\frac{2x^2}{x^2+y^2}\right ]dx+\left ( \frac{2xy}{x^2+y^2} \right )dy=0$
4 M
1(b) Solve $\left ( D^4+2D^2+1 \right )y=0$
3 M
1(c) Evaluate $\int_{0 }^{\infty }e^{-x^5}dx$
3 M
1(d) Express the following integral in polar co-ordinates: $\int_{0}^{\frac{a}{\sqrt{2}}}\int_{y}^{\sqrt{a^2-y^2}}f(x,y)dx dy$
4 M
1(e) Prove that $E=1+\Delta =e^{hD}$
3 M
1(f) Evaluate $1=\int_{0}^{\frac{\pi }{2}}\int_{\frac{\pi }{2}}^{\pi }\cos \left ( x+y \right )dx dy$
3 M

2(a) Solve $\frac{dy}{dx}+\frac{y}{x}\log y=\frac{y}{x^2}\left ( \log x \right )^2$
6 M
2(b) Change the order of integration and evaluate $1=\int_{0}^{2}\int_{\sqrt{2y}}^{2}\frac{x^2dxdy}{\sqrt{\left ( x^4-4y^2 \right )}}$
6 M
2(c) Evaluate $$\int_{0}^{\frac{\pi }{2}}\frac{dx}{1+a\sin ^2x}$$/ and duduce that $\int_{0}^{\frac{\pi }{2}}\frac{\sin ^2xdx}{\left ( 3+a\sin ^2x \right )^2}=\frac{\pi\sqrt{3} }{96}$
6 M

3(a) Evaluate $\int_{0}^{a}\int_{0}^{x}\int_{0}^{x+y}e^{x+y+2}dxdydz$
6 M
3(b) If mass per unit area varies as the square of the ordinate of a point, find the mass of a lamina bounded by the cycloid $$y=a\left ( 1-\cos \theta \right ),x=a\left ( \theta +\sin \theta \right )$$/ and the ordinates from the two cups and the tangents at the vertex.
6 M
3(c) Solve $\left ( 2x+1 \right )^2\frac{d^2y}{dx^2}-6\left ( 2x+1 \right )\frac{dy}{dx}+16y=8(2x+1)^2$
8 M

4(a) Show that the length of the are of the parabola y2=4ax cut off by the line $3y=8x \ \ \text{is}\ \ a\left [ \log 2+\frac{15}{16} \right ]$
6 M
4(b) Solve $\frac{d^3y}{dx^3}-7\frac{dy}{dx}-6y=\cos x\cosh x$
6 M
4(c) Using fourth order Runge-Kutta method, find u(0,4) of the initial value problem u'=2tu2, u(0)=1 take h =0.2.
8 M

5(a) Use method of variation of parameters to solve $\frac{d^2y}{dx^2}-5\frac{dy}{dx}+6y=e^{2x}x^2$
6 M
5(b) Using Taylor's series method, obtain the solutions of $\frac{d^y}{dx}3x+y^2, y(0)=1$ Find the value of y for x = 0.1 correct to four decimal places
6 M
5(c) Find the value of the integral $$\int_{0}^{1}\frac{x^2}{1+x^3}dx$$/ by taking h=0.2, using
i) Trapezoidal Rule ii) Simpson's 1/3 Rule. Compare errors with the exact value of the integral
8 M

6(a) A condenser of capacitance C is charged through a resistance R by a steady voltage. The charge Q satisfies the DE $$R \frac{dQ}{dt}+\frac{Q}{c}=V$$/ If the plate is chargeless find the charge and the current at time 't'
6 M
6(b) Evaluate $$\iint \frac{\left ( x^2+y^2 \right )^2}{x^2y^2}dxdy$$/ over the region common to $$x^2+y^2-ax=0$$/ and $$x^2+y^2-by=0, a>0, b>0?$$/
6 M
6(c) Find the volume common to the right circular cylinder $x^2+y^2=a^2 \text{and}\ \ x^2+z^2=a^2$
8 M

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