# Thread: Need help with a Wronskian problem.

1. ## Need help with a Wronskian problem.

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

2. Ok, so we have

$\displaystyle g'(t)-2g(t)=3e^{2t}$.

So int. factor: $\displaystyle e^{-2\int dt}=e^{-2t}.$ Multiply both sides of the equation by the int. factor to get

$\displaystyle e^{-2t}g'(t)-2e^{-2t}g(t)=\frac{d}{dt}(e^{-2t}g(t))= 3.$

Integrate both sides with respect to t to get

$\displaystyle e^{-2t}g(t)=3t+c \Longrightarrow g(t)=3te^{2t}+ce^{2t}$.

Edit: Oops! Didn't see your edit.