Picard iteration for a system

Question

I have the following system:

$$y'(t)=x^2(t)-x(t)$$

$$x'(t)=y(t)$$

It comes from the second order ode

$$x''(t)=x^2(t)'x(t)$$

I am asked to do the first four Picard iterations starting from the solution $$\phi_0 (t)= \bigg(\frac{-1}{2},0 \bigg)$$

I can do Picard iterations for a simple first order ode, but I am not able to generalize it to a system where the two equations depend on each other, and I cant find any examples or theory that tells the algorithm to help me in this case.

Answer 1

With

$$ f\left(\begin{pmatrix} y \\ x\end{pmatrix}\right) = \begin{pmatrix} x^2 - x \\ y\end{pmatrix} $$

Then $$ \phi_1(t) = \phi_0 + \int_0^t f(\phi_0(s)) \,ds \\ = \phi_0 + \int_0^t f\left(\begin{pmatrix} -\dfrac12 \\ 0\end{pmatrix}\right) \,ds \\ = \phi_0 + \int_0^t \begin{pmatrix} 0^2 - 0 \\ \dfrac34\end{pmatrix} \,ds \\ = \begin{pmatrix} -\dfrac12 \\ \dfrac34 t\end{pmatrix} $$

and so forth.