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 \(y_0=9\).
