## Applied Mathematics 2 - Dec 2012

### 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)** Using Taylor's series, find y(0.4) where dy/dx = 1 + xy with y(0) = 2.(3 marks)
**1(b)** Find the complementary function of

(3 marks)
**1(c)** Evaluate the following:

(3 marks)
**1(d)** Evaluate the following:

(3 marks)
**1(e)** Show that the following holds true:

(4 marks)
**1(f)** Using Euler's method, solve: dy/dx = x + y, y(0) =1. Find the value of y at x=1, taking h = 0.2(4 marks)
**2(a)** Evaluate the following:

(6 marks)
**2(b)** Use Runge-Kutta method of fourth order to solve dy/dx=1/(x+y); y(0)=1. Find y(0.2) with h=0.1(6 marks)
**2(c)** Solve the following:

(8 marks)
**3(a)** Solve the following:

(6 marks)
**3(b)** Solve the following using variation of parameters.

(6 marks)
**3(c)** Evaluate the following:

and hence deduce that:

(8 marks)
**4(a)** Solve the following: (xy^{3}+y)dx+2(x^{2} y^{2}+x+y^{4} )dy=0(6 marks)
**4(b)** Solve the following: (6 marks)
**4(c)** Solve the following: (8 marks)
**5(a)** In a circuit containing Inductance L, Resistance R, and Voltage E, the current i is L di/dt+Ri=E. Find i at time t. At t=0,i=0, L,R,E are constants.(6 marks)
**5(b)** Change the order of integration:

(6 marks)
**5(c)** Evaluate the following

over the volume of a sphere x^{2} + y^{2} + z^{2} = a^{2} (8 marks)
**6(a)** Find the length of the parabola x^{2} = 4y which lies inside the circle x^{2} + y^{2} = 6y.(6 marks)
**6(b)** Change into polar and evaluate: (6 marks)
**6(c)** Evaluate the following

over the area of the triangle formed by x = 0, y = 0, x + y = 1.(8 marks)
**7(a)** Change the order of integration and evaluate

(6 marks)
**7(b)** Find the area outside the circle r=a and inside the cardioid r=a(1+cosθ)(6 marks)
**7(c)** Find the volume common to the cylinders: x^{2} + y^{2} = a^{2}, x^{2} + z^{2} = a^{2}(8 marks)