I need help on how to linearize the following ODE equation so that I am able to do Stability analysis for the equation. Thanks for the help.
$\frac{dQ}{dz} = 2aM^{1/2}$
$\frac{dM}{dz} = \frac{QF}{M}$
$\frac{dF}{dz} = bQ$
Where a and b are constants. Initial points are $Q = M = 0, F = 1$ at z = 0.
Non-dimensional form of the above equation is:
$\frac{d \hat{Q}}{d\hat{z}} = \hat{M}^{1/2}$
$\hat{M}\frac{d \hat{M}}{d\hat{z}} = \hat{F}\hat{Q}$
$\frac{d \hat{F}}{d\hat{z}} = -\hat{Q}$
Case 1: \begin{equation} \begin{pmatrix} \frac{dQ}{dz} \\ \frac{dM}{dz} \\ \frac{dF}{dz} \end{pmatrix}=\begin{pmatrix} 0 & \frac{1}{M^{1/2}} & 0 \\ 0 & 0 & \frac{Q}{M} \\ -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} Q \\ M \\ F \end{pmatrix} \end{equation}
Case 2:
\begin{equation} \begin{pmatrix} \frac{dQ}{dz} \\ \frac{dM}{dz} \\ \frac{dF}{dz} \end{pmatrix}=\begin{pmatrix} 0 & \frac{1}{M^{1/2}} & 0 \\ \frac{F}{M} & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} Q \\ M \\ F \end{pmatrix} \end{equation}