For $t\ge t_0 \in \mathbb R^+$ the following recursive sequence is defined:
$$y_0(t)=y_a \\ y_{n+1}(t)=y_a+\int_{t_0}^t \alpha y_n (\tau)+\beta d\tau, \space \space \space n\in \mathbb R$$
Prove the following statement (by induction):
$$y_n(t)=(y_a+\frac{\beta}{\alpha})\sum_{k=0}^n \frac{\alpha^k}{k!}(t-t_0)^k-\frac{\beta}{\alpha}, \space \space t\ge t_0$$
First, I am not looking for a full solution. I am just looking for a some hints. I started off by showing that for $n=0$ the formula holds. Then I went on to write:
$$y_{n+1}(t)=(y_a+\frac{\beta}{\alpha})\sum_{k=0}^{n+1} \frac{\alpha^k}{k!}(t-t_0)^k-\frac{\beta}{\alpha}$$
But what does this have to be equal to? Do I need to show that
$$(y_a+\frac{\beta}{\alpha})\sum_{k=0}^{n+1} \frac{\alpha^k}{k!}(t-t_0)^k-\frac{\beta}{\alpha}=y_a+\int_{t_0}^t \alpha y_n (\tau)+\beta d\tau$$
by substituting $y_n(t)$ in the integral?