Picard's method of successive approximations

Prove that, applying Picard's method to the equation $y' = y$, with $a=0$, $b=1$, we obtain:

Prove that in the preceding example (problem 5351), $y_{n} \to y$ as $n \to \infty$ for an arbitrary choice of the function $y_{0}$.

Prove that, replacing the equation

by the system of equations

and applying Picard's method to solve these equations, we obtain a convergent process if the radius vector from the origin to the point $x$ does not pass, in the complex plane of the variable $x$, through any singular point of the functions $p(x)$ and $q(x)$.

Solve the equation $y' - y = axy$ by the method of successive approximations, setting $y_{0}=0$ and computing the quantities $y_{n}$ by the formula $y_{n}' - y_{n} = axy_{n-1}$ (at $x=0$ we must have $y=C$).

In the same way (see problem 5354), solve the equation $y'' + a^{2}y = \sigma xy$, where $\sigma$ is a small parameter. Here $y=0$, $y'=a$ at $x=0$. It is useful to apply the theorem: if $y''+a^{2}y=f(x)$, then