I am so confused in regard to sketching phase portrait for conservative systems.
For example, I have one question which says find a conserved quantity for $x''=a-e^{x}$ and sketch the phase portraits for $a \lt 0$ $a=0$ and $a \gt 0$.
I also want to ask if what I did is allowed so far;
the first part I think I did fine, it is just
$E=(1/2)x'^{2}-ax+e^{x}=$Constant
Then I tried to compute the Jacobean, yet I do not know if it is valid but I got $$J=\begin{pmatrix} 0 & 1 \\ -e^x & 0 \\ \end{pmatrix}$$
I also tried to solve for the fixed points, I got that we only have fixed points for $a \gt 0$ and they are $(ln(a),0)$ in that case the jacobian would predict a centre but since it is a borderline case I cant be sure.
So how on earth am I supposed to even begin to sketch these portraits? I really don't have much clue. Really looking for help I am very stuck here. Is there a different way in which countours can be sketched? I know I would have 3 different ones for each possible a. Should I pick values of constant and solve for points?
Here is what it should look like for the case that 'a' is negative, zero and positive respectively. but i want to understand WHY. Thanks
