A dynamic system can be represented as

dX/dt=f(X)

where X=[x1,x2,...]T is the column vector consisting of all state variables

The above equation can be linearized as

d△X=A△X

where aij in the state matrix can be calculated based on first-order taylor expansion:

aij= ∂ fi/ ∂xj | X=X0

X0 is the steady operating point of state variables.

For your question, given a system equation:

dx1/dt=x1x2;

dx2/dt=x1+x2;

then the linearized form of the first equation is

d△x1/dt=x20△x1+x10△x2

Dear Muhammad Irfan Aalam,

I completely agree with the development that Zhaoshun has done, we can stop at the level of linearization and use it as such. We can also have a model whose representation is closer to reality by using multi-model modeling using LPV model. For this I suggest you to see the following links.

https://www.sciencedirect.com/science/article/pii/S0959152412001059

https://www.inderscience.com/info/inarticle.php?artid=89621

Best regards

Hi, I agree with the comments above. Your ODE system is not linear so linearization techniques are necessary if you want to write the resulting system under the form dx/dt = A.x.