I am trying to construct a Poincaré map for:
$x’=p(t)x+q(t)$ Where $p(t) \ \text{and} \ q(t)$ are 1 periodic. I’m then asked to discuss the stability if,
$\bar{p}=\int^{1}_{0}p(s)ds$.
My first attempt was to try to solve the ODE but leaves me with an integral equation of the form,
$\dfrac{d}{dt}(x\exp(-\int p(t)dt))=q(t)\exp(-\int p(t) dt)$, but I have no idea of where to go from here.
Any help would be appreciated whether that is a different example etc that is fine.
Now integrate from $0$ to $1$ to get $$ x(1)=x(0)e^{\bar p}+\int_0^1\exp\left(\int_t^1 p(s)\,ds\right)q(t)\,dt $$ The second term is a constant wrt. $x(0)$, so that you get a discrete recursion formula $$x(n+1)=q\,x(n)+d$$ and can discuss stability based on that.
You need to be careful to not re-use variable names for different variables, here integration variables. And you need to use the same anti-derivative on both sides of the equation. I could imagine that these are stumbling blocks for you to continue.
Take $P(t)$ as a fixed anti-derivative of $p(t)$, then write $$\frac{d}{dt}(x(t)e^{-P(t)})=q(t)e^{-P(t)}$$ to make that point obvious. Then the integral can be written as $$x(t)e^{-P(t)}-x(0)e^{-P(0)}=\int_0^tq(s)e^{-P(s)}ds.$$ Note that $P(t)-P(s)=\int_s^tp(r)\,dr$, and $P(1)-P(0)=\bar p$.