General solution of differential equation ?

Question

y'+3y=t+e^(-2t)

Answer 1

It's first order linear, and so has integrating factor

e^( ∫ 3 dt) = e^(3t).

Multiply both sides by this factor:

e^(3t)(y' + 3y) = te^(3t) + e^t

(e^(3t) y)' =  te^(3t) + e^t

Integrate both sides with respect to t:

e^(3t) y = (t/3)e^(3t) - (1/9)e^(3t) + e(t) + C

y = (t/3) - (1/9) + e^(-2t) + Ce^(-3t)