## Applied Mathematics 2 - May 2013

### 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 the following:

(3 marks)
**1(b)** Solve (D^{2}-1)(D-1)^{2}y=0(3 marks)
**1(c)** Prove that E=1+Δ=e^{hD}(3 marks)
**1(d)** Solve the following:

(3 marks)
**1(e)** Change into polar co-ordinates and evaluate:

(4 marks)
**1(f)** Evaluate the following:

(4 marks)
**2(a)** Solve (x^{3}y^{3}-xy)dy=dx(6 marks)
**2(b)** Change the order of Integration and evaluate:

(6 marks)
**2(c)(1)** Prove that:

(4 marks)
**2(c)(2)** Evaluate , where a>0(4 marks)
**3(a)** Evaluate the following:

(6 marks)
**3(b)** Find the area bounded by 9xy=4 and 2x+y=2(6 marks)
**3(c)(1)** Solve the following:

(4 marks)
**3(c)(2)** Solve the equation by variation of parameters:

(4 marks)
**4(a)**

Show that for the parabola

from θ=0 to θ=π/2, length of the arc is

**4(b)**Solve the following:

(6 marks)

**4(c)**Apply Runge-Kutta method of fourth order to find an approximation value of y at x=0.2, if dy/dx=x+y

^{2}, given y=1 when x=0, in steps of h=0.1(8 marks)

**5(a)**Solve: (2xy

^{4}e

^{y}+2xy

^{3}+y)dx+(x

^{2}y

^{4}e

^{y}-x

^{2}y

^{2}-3x)dy=0(6 marks)

**5(b)**Solve dy/dx=2x+y with x

_{0}=0,y

_{0}=0 by Taylor’s method. Obtain y as a series in power of x. Find approximation value of y for x=0.2,0.4. Compare your result with exact values.(6 marks)

**5(c)**Evaluate the following equation

by Trapezoidal method, Simpson's 1/3rd and 3/8th methods. Compare result with exact values.(8 marks)

**6(a)**In a circuit containing inductance L, resistance R, voltage E, the current i is given by L(di/dt)+Ri=E. Find i at a time t if at t=0,i=0, and if L, R, E are constants.(6 marks)

**6(b)**Evaluate ∫∫xy dxdy bounded by y=x, x

^{2}+y

^{2}-2x=0, and y

^{2}=2x(6 marks)

**6(c)(1)**Find the volume of a tetrahedron bounded by the plane x=0,y=0,z=0 and x+y+z=a(4 marks)

**6(c)(2)**Find the volume bounded by the cone z

^{2}=x

^{2}+y

^{2}and paraboloid z=x

^{2}+y

^{2}.(4 marks)