solve the following differential eqaution for $\gamma\left(t\right)=e^{-\alpha t } (\alpha > 0) $
$\frac{d \phi(t)}{d t}=-\int_0^t \gamma\left(t-t^{\prime}\right) \phi\left(t^{\prime}\right) d t^{\prime}$ , $\phi\left(0\right)=\phi_0 $
Can someone please kindly help me? Also is there any specific name for this kind of differential equation problem which I can use for google search keyword to try to find some solution?
Differentiating both sides with respect to $t$, one can transform the integro-differential equation $$ \frac{d }{d t}\phi(t)=-\int_0^t e^{-\alpha(t-t')} \phi(t')\,d t' \tag{1} $$ into an ODE: \begin{align} \frac{d^2 }{d t^2}\phi(t)&=-\phi(t)+\alpha\int_0^t e^{-\alpha(t-t')} \phi(t')\,dt' \\ &=-\phi(t)-\alpha\frac{d }{d t}\phi(t) \tag{2}. \end{align} Since $(2)$ is a second order ODE, one needs an additional constant of integration. It can be obtained by taking $t=0$ in $(1)$: $$ \phi'(0)=0. \tag{3} $$ The solution of $(2)$ is left as an exercise.