If the Wroian W of f and g is 3e4t, and if (t)=e2t, find g(t).

${\displaystyle W(e^{2t},g\left(t\right))=e^{2t}{g}^{\prime }\left(t\right)-2e^{2t}g\left(t\right)=3e^{4t}}$, divide both sides with ${\displaystyle e^{2t}}$
${\displaystyle g^{\prime }-2g=3e^{2t}}$
This is first order linear equation. Find integrating factor:
${\displaystyle \mu \left(t\right)=\exp (\int -2dt)=e^{-2t}}$
Multiply equation with $\mu$ :
${\displaystyle e^{-2t}{g}^{\prime }-2e^{-2t}g=3}$
${\displaystyle {\left(e^{-2t}g\right)}^{\prime }=3}$, integrate
${\displaystyle e^{-2t}g=3t+c}$
${\displaystyle g=3t{e}^{2t}+c{e}^{2t}}$
Result:
${\displaystyle g=3t{e}^{2t}+c{e}^{2t}}$