I'm confident i've got the correct ODE. (part a)

\(\frac{di}{dt} + \frac{1}{RC}i = \frac{d{V}_{s}}{dt}\frac{1}{R}\)

I come unstuck with part b

I'm not sure how to solve it with a \({V}_{s}(t) = 12t\)

If I use the non homogeneous methord to solve it,

I get a \({y}_{H} = i(t) ={C}_{1}{e}^{-\frac{1}{RC}t}\)

and then I don't know what my guess is for the RHS = \(\frac{d{V}_{s}}{dt}\frac{1}{R}\)

have I got the incorrect ODE in the first place?

any help is greatly appreciated!!