## Applied Mathematics 2 - May 2014

### 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)** Evaluate $\displaystyle \int_{0}^{1}\sqrt{(x-x)}dx$(3 marks)
**1(b)** Solve D^{4}-4D^{3}-8D+4]y=0(3 marks)
**1(c)** Prove that (1+Δ)(1-∇)=1(3 marks)
**1(d)** Change to polar co-ordinates and evaluate $\displaystyle \int_{0}^{a}\int_{0}^{\sqrt{x^{2}-x^{2}}}dydx$(3 marks)
**1(e)** Solve(X^{4}-4xy -2y^{2})dx+(y^{4}-4xy -2x^{2})dy =0
(4 marks)
**1(f)** Evaluate $$\int_{0}^{2}\int_{0}^{\sqrt{1+x^{2}}}\dfrac{1}{1-x^{2}-y^{2}}dy\ dx$$(4 marks)
**2(a)** Solve $$xy \left(1+xy^{2}\dfrac{dy}{dx}\right)=1$$(6 marks)
**2(b)** Change the order of integration and evaluate $$\int_{0}^{\alpha}xe^{-x^{2}/2}dy\ dx $$(6 marks)
**2(c)** Evaluate $$\int_{0}^{\pi/2}\dfrac{dx}{a^{2}sin^{2}x+b^{2}cos^{2}x}$$ and show that

$$\int_{0}^{\pi/2}\dfrac{dx}{(a^{2}sin^{2}x+b^{2}cos^{2}x)^{2}}=\dfrac{pi}{4ab}\left(\dfrac{1}{a^{2}+\dfrac{1}{b^{2}}}\right)$$(8 marks)
**3(a)** Evaluate $$\iiint x^{2}yz\ dx\ dx\ dz$$ through the volume bounded by x=0, y=0, z=0. x+y+z =1(6 marks)
**3(b)** Find the area bounded by parabola y^{2}=4x and the line y = 2x-4.(6 marks)
**3(c)** Use the method of variation of parameter to solve

$$\dfrac{d^{2}y}{dx^{2}} + 5\dfrac{dy}{dx}+6y =e^{-2x}\sec^{2} x+1 (1+2\tan x)$$(8 marks)
**4(a)** find the total length of the loop of the curve $$9y^{2}=(x+7)(x+4)^{2}$$(6 marks)
**4(b)** Solve $$\dfrac{d^{2}y}{dx^{2}}+2y -x^{2}e^{3x}+e^{x}-\cos 2x$$(6 marks)
**4(c)** Apply Runge-kutta method of fourth order to find an approximate value of y at x=0.2 if $$\dfrac{dy}{dx}=x+y^{2} $$ given that y=1 when x=0 in step of h=0.1(8 marks)
**5(a)** Solve y(X^{2}y + e^{x})dx-e^{x}dy =0.(6 marks)
**5(b)** using taylor's series method solve $$\dfrac{dy}{dx}=1-2xy$$ given that y(0)=0 and hence y=(0,2) and y(0,4).(6 marks)
**5(c)** Compute the value of the definite integral $$\int_{0.2}^{1.4}(\sin x-log_{e}x+e^{x})dx$$, by

Trapezoidal Rule

Simpson's one third Rule

Simpson's three-eighth Rule.(8 marks)
**6(a)** The Motion of a particle is given by $$\dfrac{d^{2}x}{dt^{2}}=-k^{2}x-2h\dfrac{dx}{dt}$$solve the equation when h=5, k=4 taking x=0, v=v_{0} at t =0. Show that the the time of maximum displacement is independent of the initial velocity.(6 marks)
**6(b)** Evaluate$$ \iint(x^{2}+y^{2})dx\ dy $$ over the area of triangle whose vertices are (0,0),(1,0)(1,2).(6 marks)
**6(c)** Find the volume bounded by y^{2} =x, x^{2} =y and the planes z=0 and x+y+z=1.(8 marks)