Calculate the flow of the differential equation $x''+ax'+1=0$ for every value of the parameter $a$. Discuss the behavior of the flow as a function of the parameter. Hint: Transform to a linear system. Find the eigenvalues of the matrix to study the system through its Jordan normal form. Be careful with the multiplicity of the eigenvalues.
While I would have no problem with the Jordan decomposition, I'm having issues in transforming the differential equation in a linear system of the form $x'=Ax$ for a square matrix $A$, since we've studied up until now just this form and not the case $x'=Ax+b$ for a vector $b$.
But my attempt stops in the moment where I find a matrix, but it is necessary to add a vector $b$, for the system to hold, e.g.
Let $y=\begin{pmatrix}y_1 \\\ y_2\end{pmatrix}=\begin{pmatrix}x\\\ x'\end{pmatrix}$ s.t. $y'=\begin{pmatrix}x'\\\ x''\end{pmatrix}$, we have $y_2=y_1'=x'$, $y_2'=-ay_2-1$
Then $\begin{pmatrix}y_1'\\\ y_2'\end{pmatrix}=\begin{pmatrix}y_2\\\ -ay_2-1\end{pmatrix}=\begin{pmatrix}0&1\\\ 0&-a\end{pmatrix}\begin{pmatrix}y_1\\\ y_2\end{pmatrix}+\begin{pmatrix}0\\\ -1\end{pmatrix}$, i.e. in the form $y'=Ay+b$ where $A=\begin{pmatrix}0&1\\\ 0&-a\end{pmatrix}$ and $b=\begin{pmatrix}0\\\ -1\end{pmatrix}$.
Am I missing something? Any help is appreciated.