$$ \begin{cases} \frac{dx_1}{dt} = 0.25 x_1 - 0.001 x_1 x_2 - 2.5 \cdot (10^{-5}) x_1^2 \\ \frac{dx_2}{dt} = -0.5x_2 + 0.001 x_1 x_2 - 5 \cdot 10^{-4} x_2^2 \end{cases} $$ Initial conditions are $x_1(0) = 500$; $ x_2(0)=50$, if that's any help.
I need to transform this system into the form: dx/dt = f(x), where dx and f(x) are vectors.
I've read that I need to do Jacobian matrix, but I don't know what to do with it after I compute it.
Any help and ideas will be much appreciated. Thank you.
Assuming that by "x" you mean the vector $\begin{pmatrix}x_1 \\ x_2 \end{pmatrix}$ then your vector form is simply $\frac{dx}{dt}= \begin{pmatrix}\frac{dx_1}{dt} \\ \frac{dx_2}{dt} \end{pmatrix}= \begin{pmatrix}0.25x_1- 0.001x_1x_2- 2.5(10^{-5})x_1^2 \\ -0.5x_1+ 0.001x_1x_2- 5(10^{-4})x_2^2 \end{pmatrix}$