If the Wronskian W of f and g is $\displaystyle 3e^{4t}$, and if $\displaystyle f(t

)=e^{2t}$ find g(t).

I know we get $\displaystyle e^{2t}g'(t)-2e^{2t}g(t)=3e^{4t}$ which becomes $\displaystyle g'(t)-2g(t)=3e^{2t}$.

It is after this that I am getting messed up. I know you use the integrating factor method to solve this, but I am not coming up with the correct answer shown in my book which is $\displaystyle g(t)=3te^{2t}+ce^{2t}$ I know how to solve first order linear questions but apparently I'm doing something wrong. Can someone help please?

EDIT: I don't need help with this anymore! I realized the mistake I was making