Solve the equation for θ given that when t = 0, θ = 0 and dθ

dt = 0. (12)

3. Determine y(n) when y = 2x3e4x (10) 4. Determine the power series solution of the dif-

ferential equation: d2y dx2

+ 2x dy dx

+ y = 0 using Leibniz-Maclaurin’s method, given the boundary conditions that at x = 0, y = 2 and dy

dx = 1. (20)

5. Use the Frobenius method to determine the general power series solution of the differential

equation: d2y dx2

+ 4y = 0 (21)

6. Determine the general power series solution of Bessel’s equation:

x2 d2y dx2

+ x dy dx

+ (x2 − v2)y = 0

and hence state the series up to and including the term in x6 when v=+3. (26)

7. Determine the general solution of ∂u

∂x = 5xy

(2)

8. Solve the differential equation ∂2u

∂x2 = x2(y − 3)

given the boundary conditions that at x = 0, ∂u

∂x = sin y and u = cos y. (6)

9. Figure A14.1 shows a stretched string of length 40 cm which is set oscillating by displacing its mid-point a distance of 1 cm from its rest position and releasing it with zero velocity. Solve the

wave equation: ∂2u

∂x2 = 1

c2 ∂2u

∂t2 where c2 = 1, to

determine the resulting motion u(x, t). (23)