I am asked the following:
Consider the following planar dynamical system, where $A \in R$ is a parameter:
$$d/dt(x,y) = (-x,x^2+sinh(x)(A-y))$$
a) Find all equilibria. Your answer will depend on A.
Here is what I have done so far:
Recall that $$sinh(x) = \frac{e^x-e^{-x}}{2}=...=\sum_{k=1}^{\infty} \frac{x^{2k}}{(2k)!}$$
Letting $\dot{y}=\dot{x}=0$, we get $x=y=0$. The other equilibrium I find is
$$x^2+\frac{x^2}{2}(A-y)=0$$ $$\Rightarrow x^2+\sum_{k=1}^{\infty} \frac{x^{2k}}{(2k)!}(\frac{A-y}{2})=0$$ $$\Rightarrow 1+\sum_{k=1}^{\infty} \frac{x^{2k}}{(2k)!}(\frac{A-y}{2})=0$$ $$\Rightarrow 2+\sum_{k=1}^{\infty} \frac{x^{2k}}{(2k)!}(A-y)=0$$
Call $f(x)=\sum_{k=1}^{\infty} \frac{x^{2k}}{(2k)!}$.
This gives $$x^2 + \frac{x^2}{f(x)}(A-y) = 0$$. Solving for $y$, I only seem to get one equilibrium point, $(x,y)=(0,A)$. But it says equilibria, are there more than one?
The second equilibrium feels weird, because it looks like I get 0=0, and I can't find a defined x given $y=f(A)$.