Determine a sinusoidal expression for the resultant and

Question

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 v_R = v₁ + v₂ by finding the horizontal and vertical components.

ii. Determine the resultant v_R = 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

0	2	1
4	1	0
4	0	1

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 J_v (x) and J_(-v) (x) as:

AJ_v(x)+ (BJ)_(-v) (x) = (x/y)^v {1/(τ(v+1)) - x²/(2² (1!)τ(v+2)) + x⁴/(2⁴ (2!)τ(v+4))- ....}