6 (a) Find the solution to the differential equation

dy

dt

+6y= 8e

subject to the condition that y = 1 when t = 0.

[3 marks]

(b) The linear differential equation (with non-constant coefficients)

t2d²y

dy

dt

7t- + 15y = 0

dt²

(1)

has homogeneous solution y = Pt³+ Qt5. Now consider the non-

homogeneous equation

2d²y

dt²

7t

dy

dt

+ 15y = 3t² - 32t

(2)

i. Find the general solution to the differential equation (2)

ii. Find the particular solution satisfying the differential equation (2)

together with the conditions that y = 3 and

24 when t = 1.

iii. Explain why it would not be possible to satisfy conditions for

equation (2) at t = 0.

dy

dt