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(\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.