Question

Find the general solution to the coupled system

dx₁ /dt = 2x₁ +x₂

dx₂/dt = x₁ + 2x₂

Sketch the phase portrait for the system and classify the origin as a node, a saddle, a center, or a spiral. Is the origin unstable, asymptotically stable, or stable? Explain and show as much work as needed

Answer 1

The equations given to us are first order linear homogenous differential equations. So hopefully they won't be too difficult to solve. Writing them down.

Alright, we can write them in a matrix form, such that,

Substituting what is given to us,

For the matrix A, we find the eigen values using the characteristic equation,

From this, we need to find eigen vectors for both eigen values. Let us name them

For,

From the above two equations it is clear that

Therefore, the eigen vector is,

Similarly we solve for the other eigen value,

Again it is clear that,

Hence, the eigen vector is given by,

Now, for a first order solution the general form is,

We also know that a linear solution can be written as a sum of other linear solutions, Therefore our combined general solution is,

Subsituting we have two final equations,

The above equations are the general solutions to our system of equations.

Now, check out the given equations. Here, only when . This means the origin is the only critical point of the system. So our solutions in the phase plane will either move away or towards the critical point. What are phase planes anyway?

Well, we are a coupled system. This means we have a system where, here for instance, the time derivative of one variable is dependent on the other variable. and similarly for the time derivative of the other one, .

This interconnectedness makes it difficult to find a general solution. But we use the property of linear systems that the sum of linear solutions is also a linear solution and hence, our general linear solution can be written as a sum of independent linear solutions.

We use the matrix notation to do that. Once we have done it, we get a set of solutions that will satisfy this coupled system. Since, both variables are parameterized in time, we can plot a graph of the solutions in a cartesian system. When we do that we get how the base variables of the system move with respect to each other at any time . This 'plot' is called a phase portrait. The cartesian system is called the phase plane, and the path of the system, is called the trajectory. Sketching these without electronic help is very cumbersome but a rough idea of the portrait is very much possible.

To sketch the phase portrait, we write down our eigen vectors first,

It is an unstable improper node. The phase portrait is,