Sketch the phase line of ODE: $\frac{dx}{dt}=f(t,x)$

Question

I was asked to sketch the phase line of scalar ODE, and using the phase lines, determine whether all equilibrium point(s) are sinks, sources, or nodes. Here is the scalar ODE: $$\frac{dx}{dt}=f(t,x)$$ where

$f(t,x):=\frac{2-π}{π}$, if $x\ge \frac{π}{2}$

$f(t,x):=\frac{sinx}{x}-1$, if $0\lt x\lt \frac{π}{2}$

$f(t,x):=xe^x$, if $x\le 0$

For $x\le 0$, I calculated $f(t,x)=0$ and got $x=0$ as the equilibrium point.

For $0\lt x\lt \frac{π}{2}$, I have no idea how to calculate and proceed on $f(t,x)=0$ as I would result in $sinx=x$.

For $x\ge \frac{π}{2}$, I don't think there exists any equilibrium points.

I do know how to distinguish between sinks, sources, and nodes, but I do not have a clear picture on way to determine all the equilibrium points. Thank you.

Answer 1

Here is the stream plot of your function:

---

In Mathematica:

f[t_, x_] := 
  Which[x <= 0, x Exp[x], 0 < x <= \[Pi]/2, Sin[x]/x - 1, 
   x > \[Pi]/2, (2 - \[Pi])/\[Pi]];
StreamPlot[
 Evaluate[D[
   f[t, x], {{t, x}}]], {x, -\[Pi], \[Pi]}, {t, -\[Pi], \[Pi]}, 
 StreamScale -> Medium]