TASK 1:
a. The two alternating voltages are given by v1 = 15 sinωt volts and v2 = 25 sin?(ωt-π/6)volts.
i. Determine a sinusoidal expression for the resultant vR = v1 + v2 by finding the horizontal and vertical components.
ii. Determine the resultant vR = v1 - v2 using horizontal and vertical components.
(b) Calculate the 1st and 2nd moment of area for the shape shown about the axis s-s and find the position of the centroid.
(c) Find the eigenvalues and eigen vectors for the matrix
TASK: 2:
Determine the power series solution of the differential equation:
(d2y)/dx2 +xdy/dx + 2y = 0 Using Leibniz - Maclaurin's method, given the boundary conditions that at x = 0, y = 1 and dy/dx = 2
TASK 3:
Determine the general power series solution of Bessel's equation.
x2(d2y)/dx2 +x.dy/dx + (x2 - v2) y = 0
TASK 4:
Show that the power series solution of the Bessel equation of the above problem may be written in terms of the Bessel functions Jv (x) and J(-v) (x) as:
AJv(x)+ (BJ)(-v) (x) = (x/y)v {1/(τ(v+1)) - x2/(22 (1!)τ(v+2)) + x4/(24 (2!)τ(v+4))- ....}