I have a really simple separable DE from an input output problem and I'm having trouble with the constant. (I think)

and

let then

So I'm guessing that my problem starts here. I know I have to exponentiate both sides to solve for the dependant variable. And I'm assuming that after exponentiation the equation looks like this;

Even if I were to evaluate the problem with the initial value provided this doesn't seem to match the provided answer, , at all.