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: (xy³+y)dx+2(x² y²+x+y⁴ )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 x² + y² + z² = a² (8 marks) 6(a) Find the length of the parabola x² = 4y which lies inside the circle x² + y² = 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: x² + y² = a², x² + z² = a²(8 marks)