91Ó°ÊÓ

The phase plane is a graphical representation of the behavior of dynamic systems. In the context of a system of differential equations, it provides insight into the system's trajectories over time.
By plotting variables

• typically one for each axis,
• such as the dependent variables in a set of first-order ODEs,

we gain understanding about their interplay.
In our exercise, the phase plane helps visualize how the system of equations evolves. Specifically, with the two equations given by

• \(y_1' = y_2\) and
• \(y_2' = 4y_1 - y_1^3\),

we can draw trajectories based on the initial values of the variables. The x-axis in the phase plane corresponds to \(y_1\) while the y-axis corresponds to \(y_2\).
Various initial conditions will result in different trajectories, showcasing stable and unstable points. Stable points attract trajectories, whereas unstable points repel them.

A system of equations consists of multiple equations dealing with multiple variables. When considering differential equations, we often convert higher-order ODEs into a system of first-order ODEs to simplify the solution process.
Given the second-order differential equation: \[ y^{\prime \prime}-4 y+y^{3}=0 \]We set:

• \(y_1 = y\)
• \(y_2 = y'\)

This results in the system:

• \(y_1' = y_2\)
• \(y_2' = 4y_1 - y_1^3\)

The conversion to a system of equations allows us to handle the problem using linear algebra techniques and numerical methods that are more straightforward with first-order systems. The general idea is to express each derivative

• like \(y_1\) and \(y_2\)

in terms of each other or constants, forming a comprehensible and solvable structure.

Finding solutions to ordinary differential equations (ODEs) often involves understanding various methods, whether they be analytical or numerical.
For the system of equations:

• \(y_1' = y_2\)
• \(y_2' = 4y_1 - y_1^3\)

We look for solutions that describe \(y_1\) and \(y_2\)'s behavior over time.
While exact, closed-form solutions might not always be possible, we can analyze the solution structure for particular initial conditions. In our exercise, we note that \(y_2\)'s behavior depends on initial conditions and the trajectory of \(y_1\).
Numerical methods such as Euler's method or Runge-Kutta methods can provide approximate solutions by incrementally progressing through time steps, using known initial values.