Applied Mathematics 2 - Dec 2016

First Year Engineering (Semester 2)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 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 marks) 1(b) Solve $$\left ( D^4+2D^2+1 \right )y=0$$(3 marks) 1(c) Evaluate $$\int_{0 }^{\infty }e^{-x^5}dx$$(3 marks) 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 marks) 1(e) Prove that $$E=1+\Delta =e^{hD}$$(3 marks) 1(f) Evaluate $$1=\int_{0}^{\frac{\pi }{2}}\int_{\frac{\pi }{2}}^{\pi }\cos \left ( x+y \right )dx dy$$(3 marks) 2(a) Solve $$\frac{dy}{dx}+\frac{y}{x}\log y=\frac{y}{x^2}\left ( \log x \right )^2$$(6 marks) 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 marks) 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 marks) 3(a) Evaluate $$\int_{0}^{a}\int_{0}^{x}\int_{0}^{x+y}e^{x+y+2}dxdydz$$(6 marks) 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 marks) 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 marks) 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 marks) 4(b) Solve $$\frac{d^3y}{dx^3}-7\frac{dy}{dx}-6y=\cos x\cosh x$$(6 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks)