Integration of a recursive function

Question

I have the following equation (where $\alpha$ is a constant and $t \in \mathbb R$): $$f_k'=\alpha f_{k-1}(t)$$ whose solution has to be: $$f_k(t)=\frac{(\alpha t)^k}{k!}$$ But how?

Answer 1

Ok. Let's start with $f_0$ and assume that the initial condition is $0$ (i.e. $f_0(0) = 0$). We know that $(f_1)'(t) = \alpha f_0$.

Then, $f_1(t) = \alpha t f_0$. Next, we know that $(f_2)'(t) = \alpha f_1(t) = \alpha ^2 tf_0$. Integrating, we find that $f_2(t) = \frac{1}{2} \alpha^2 t^2 = \frac{1}{2} (\alpha t)^2$. Can you finish the problem from here?