Question

Uppose y(t)=9e−5t

Answer 1

If $y(t)=9e^{-5t}$ satisfies the ODE $y'-ky=0$, then

$y'(t)=-45e^{-5t}$

and substituting into the ODE gives

$-45e^{-5t}-9ke^{-5t}=0\implies k=-5$

so that the ODE is $y'+5y=0$.

I'm assuming that you're also supposed to determine the initial value $y_0$ given this ODE. You have

$y'+5y=0\implies e^{5t}y'+5e^{5t}y=0\implies(e^{5t}y)'=0\implies e^{5t}y=C\implies y=Ce^{-5t}$

Given $y(0)=y_0$, you have

$y_0=C$

but since you already know that $y=9e^{-5t}$, it follows that $C=9$, and in turn that $y_0=9$.