Find the general solution of the differential equation?

Question

dy/dt - e^-3t + 4y= 2t-1

Answer 1

Cast this as:
(D+4)y = e^(-3t) + 2t - 1, where D = d/dt

So now we pick off components:
Assume a solution of the form y = Ce^(-3t)
Then
(D+4)y = -3Ce^(-3t) +4Ce^(-3t) = Ce^(-3t) so that C = 1

Now assume a solution of the form y = Et + F
Then
(D+4)y = E + 4Et + 4F = 2t - 1
so that E = 1/2 and F = -3/8

Finally, if y is of the form Ge^(-4t) then
(D+4)y = 0

So the general solution is:
y = e^(-3t) + t/2 -3/8 + Ge^(-4t)