Question

Prove that the origin is a stable node for x′ = −x, y′ =

−2y.
a. Through its explicit solution
b. Finding the orbits in the phase plane

Answer 1

To prove that the origin is a stable node for the system of differential equations x′ = −x and y′ = −2y, we can use two methods: finding the explicit solution and finding the orbits in the phase plane.

Step-by-step explanation:

To prove that the origin is a stable node for the system of differential equations x′ = −x and y′ = −2y, we can use two methods.

In method (a), we find the explicit solution.

The solution is x = Ae^(-t) and y = Be^(-2t), where A and B are constants.

In method (b), we can find the orbits in the phase plane.

By graphing the vector field of the system and analyzing the behavior of the trajectories, we can see that all trajectories approach the origin as t approaches infinity, confirming that the origin is a stable node.