Question

If thewronskian w of f and g is 3e4t,and if f(t) = e2t,find g(t).

Answer 1

$W(f(x),g(x))=\begin{vmatrix}f(x)&g(x)\\f'(x)&g'(x)\end{vmatrix}=f(x)g'(x)-g(x)f'(x)$

We have $f(t)=e^{2t}\implies f'(t)=2e^{2t}$, so

$W(f(t),g(t))=e^{2t}g'(t)-2e^{2t}g(t)=3e^{4t}$
$\implies e^{-2t}g'(t)-2e^{-2t}g(t)=3$
$\implies\dfrac{\mathrm d}{\mathrm dt}[e^{-2t}g(t)]=3$
$\implies e^{-2t}g(t)=\displaystyle\int3\,\mathrm dt$
$\implies e^{-2t}g(t)=3t+C$
$\implies g(t)=3te^{2t}+Ce^{2t}$

where $C$ is any arbitrary constant.