If the Wronskian W of f and g is , and if find g(t).

I know we get which becomes .

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 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