Consider the following difference-differential equation $$ (1+\alpha t)y_k'(t) - (k\beta+r) y_k(t) = y_{k-1}(t) $$ with $\alpha,\beta,r\in\mathbb R$ and the initial conditions $y_k(0)=0$ for $k\geq1$ and $y_0(t)=(1+\alpha t)^{r/\alpha}$.
In a paper I'm currently reading it is said, that this differential equation has a unique solution under the above conditions and I'm wondering why. If I could rewrite the equation to $y_k'(t)=f(t,y_k)$ I could probably use the existence and uniqueness theorem, unfortunately the equation also contains $y_{k-1}(t)$ so I can just rewrite the equation as $$ y_k'(t) = f(t,y_k(t),y_{k-1}(t)). $$ How can I show that there exists a unique solution to this type of differential equation?