It is given differential equation $x'=x^3-x\mu$, $x\in \mathbb R$ with parameter $\mu \in \mathbb R $.
i) Can initial value problem $x(0)=a$,$a \in \mathbb R$ be solved explicitly? Explain.
ii) Determine all rest points and their type? Make sketch in $(\mu,x)$ plane where on vertical lines rest points $\mu=const$ are represented and sketch their behavior with arrows.
i) I don't know. Can someone give me some hint?
ii) $f(x)=x^3-x\mu$
$f(x)=0$
$x_1=0,x_2=\mu,x_3=-\mu$
$f'(0)=-\mu<0 \mapsto$ attractive rest points
$f'(\mu)=3\mu^2-\mu>0 \mapsto $ repulsive rest points
$f'(-\mu)=3\mu^2-\mu>0 \mapsto$ repulsive rest points
Sketch idea:
Is this correct and if yes, how to add "arrows"?
P.S. Repulsive/attractive rest points isn't probably the best translation, but I didn't know how to translate it right. On german it is: "abstossend/anziehend Ruhelagen".
i) If $\mu > 0$,
$$ x(t) = \frac{a \sqrt{\mu}}{\sqrt{a^2+(\mu-a^2) \exp(2 \mu t)}}$$
(I'll leave the cases $\mu=0$ and $\mu < 0$ to you).
ii) No. $x^3 - x \mu = x (x^2 - \mu) = x (x-\sqrt{\mu})(x+\sqrt{\mu})$. So $x = 0$ or (if $\mu > 0$) $x = \pm \sqrt{\mu}$.