[tex]W(f(x),g(x))=\begin{vmatrix}f(x)&g(x)\\f'(x)&g'(x)\end{vmatrix}=f(x)g'(x)-g(x)f'(x)[/tex]
We have [tex]f(t)=e^{2t}\implies f'(t)=2e^{2t}[/tex], so
[tex]W(f(t),g(t))=e^{2t}g'(t)-2e^{2t}g(t)=3e^{4t}[/tex]
[tex]\implies e^{-2t}g'(t)-2e^{-2t}g(t)=3[/tex]
[tex]\implies\dfrac{\mathrm d}{\mathrm dt}[e^{-2t}g(t)]=3[/tex]
[tex]\implies e^{-2t}g(t)=\displaystyle\int3\,\mathrm dt[/tex]
[tex]\implies e^{-2t}g(t)=3t+C[/tex]
[tex]\implies g(t)=3te^{2t}+Ce^{2t}[/tex]
where [tex]C[/tex] is any arbitrary constant.
