Questions about the Picard–Lindelöf theorem for the ODE $y(0) = y_0 (1)$ $y'(x) = αy(x)$

Question

Hi i have questions about an exercise that we have done in class:

We consider the following ODE for a given $y_0 ∈ \mathbb{R}$ and $α ∈ \mathbb{R}$ $$ \left.\begin{gathered} y(0) = y_0 \\ y'(x) = αy(x) \end{gathered}\right]\tag1 $$ This problem fulfills the requirements for the theorem of Picard-Lindelöf. According to the associated proof the solution to the problem, $$ y(x_0) = y_0, \\ y'(x) = f(x, y(x)), $$ where $f$ is Lipschitz continuous with respect to the second variable, can be iteratively calculated by the so called Picard-Iteration $$ϕ_{k+1}(x) = y_0+\int_{x_0}^{x} f(s, ϕ_k(s)) ds$$

for $k ∈ \mathbb{N}_0$ with $ϕ_0(x) ≡ y_0$. The Picard-Iteration then converges uniform to the solution of the presented problem.

Solve (1) by application of the Picard-Iteration.

So what we have done is the following: We know that $ϕ_0(x)≡y_0$

Then we have $$ϕ_1(x)=y_0+\int_{0}^{x}\alpha y_0ds\tag2=y_0(1+\alpha x)$$ so we have for $$ϕ_{n+1}(x)=y_0+\int_{0}^{x}\alpha y_0 \sum_{l=0}^{n}\frac{(\alpha s)^l}{l!}ds \tag3 \\ ... \\ y(x)=\lim_{n\rightarrow \infty}ϕ_n(x)=y_o\sum_{l=0}^{\infty}\frac{(\alpha x)^l}{l!}=y_0e^{\alpha x}$$

My first question is: by (2) how can we know that the Picard iteration $$ϕ_{k+1}(t)=y_0+\int_{t_0}^{t}f(s,ϕ_k(s))ds$$ for $k+1=1$ is actually $$ϕ_1(x)=y_0+\int_{0}^{x}\alpha y_0ds$$ or more concretely, how do we know $f(s,ϕ_1(s))=\alpha y_0$?

My second question is by (3). How do we arrive at this equation? It's not clear to me where the $\alpha y_0 \sum_{l=0}^{n}\frac{(\alpha s)^l}{l!}$ comes from. Can someone help me?

EDIT: my first question is solved

Answer 1

As you iteratively calculate the integral of a polynomial and always add a constant term, you get a power series in the limit. As you, when integrating, always again apply $f$, which is multiplying by $\alpha$, you also get the same power of $\alpha$ as of your variable. Then because you integrate a power of your variable, you have to divide by the power + 1, because you do this for every power, so every natural number, you get the factorial. This power series equals the exponential function, may that be by definition or proof.

This should make it clear, when you write down the equations yourself.