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MU First Year Engineering (Semester 2)
Applied Mathematics 2
May 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) Prove that $\int ^1_0\frac{dx}{\sqrt{-\log x}}=\sqrt{\pi}$
3 M
1(b) Solve $\dfrac{d^3y}{dx^3}-5\dfrac{d^2y}{dx^2}+8\dfrac{dy}{dx}-4y=0$
3 M
1(c) Prove that Δ∇=Δ&nabla
3 M
1(d) Solve [xy sin(xy) + cos (xy)] y dx+[x ysin(xy) - cos (xy)]x dy=0
3 M
1(e) Change to polar coordinates and evaluate $\displaystyle \int ^1_0\int ^{\sqrt{2x-x^2}}_x \left ( x^2+y^2 \right )dx\ dy$
4 M
1(f) Evaluate $\int ^1_0\int ^x_0\left ( x^2+y^2 \right )xdy\ dx$
4 M

2(a) Solve $\left ( 1+y^2 \right )dx=\left ( e^{\tan^{-1}y}-x \right )dy$
6 M
2(b) Change the order of integration and evaluate $\int ^1_0\int ^{\sqrt{1-x^2}}_0\frac{e^y}{\left ( e^y+1 \right )\sqrt{1-x^2-y^2}}dy\ dx$
6 M
2(c) Prove that $\int ^{\infty}_0\dfrac{e^{-x}-e^{-ax}}{xsecx}dx=\dfrac{1}{2}\log\left ( \dfrac{a^2+1}{2} \right )$
8 M

3(a) Evaluate $\int ^e_1\int ^{\log y}_1\int ^{e^x}_1\log z\ dz\ dy\ dx$
6 M
3(b) Find the total area od the curve r = a sin2?
6 M
3(c) Solve $x^2\dfrac{d^3y}{dx^3}+3x\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}=x^2\log x$
8 M

4(a) Show that thw length of the arc of the curve ay2 = x3 from the origin to the point whose abscissa is b is $\dfrac{8a}{27}\left [ \left ( 1+\dfrac{9b}{4a} \right )^{3/2} -1\right ]$
6 M
4(b) Solve $$\left ( D^2-D-2 \right )y=2 \log x+\dfrac{1}{x}+\dfrac{1}{x^2}$$
6 M
4(c) Apply Runge-Kutta Method of fourth order to find an approximate value of y for $$\dfrac{dy}{dx}=x\ y$$ with x0=1, y0 =1 at x=1.2 taking h=0.1
8 M

5(a) Solve $$\left ( x^2y-2xy^2 \right )dx-(x^3-3x^2y)dy=0$$
6 M
5(b) Using Taylor series Method obtain the solution of following differential equation $$\dfrac{dy}{dx}=2y+3e^x$$ with y0 = 0 when x0 = 0 for x=0.1, 0.2
6 M
5(c) Find the approximate value of $$\displaystyle \int ^4_0 e^x dx$$
by i) Trapezoidal Rule, ii) Simpson's 1/3rd Rule
8 M

6(a) In a circuit containing inductance L, resistance R, and voltage E, the current I is given by $$L\dfrac{di}{dt}+Ri=E.$$ Find the current i at time t if at t=0, i=0 and L, R, E are constants.
6 M
6(b) Evaluate $$\iint _R\dfrac{dxdy}{\left ( 1+x^2+y^2 \right )^2}$$ over one loop of the lemniscate $$\left ( x^2+y^2 \right )^2=x^2-y^2$$
6 M
6(c) Find the volume bounded by the cylinder x2 + y2 = 4 and the planes z=0 and y+z=4
8 M

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