\begin{equation} y’ + ay = f(t) \end{equation}

The general solution for this would be

1. any solution specifically which gives \(f(t)\), plus
2. any homogeneous solutions

specifically:

\begin{equation} y = y_{p}(t) + y_{n}(t) \end{equation}

where the left is a particular solution, and the right is any homogeneous solution. We can do this because, say if we derivate it; the left derivative (the particular solution) gives \(f(t)\), and the right, because its homogeneous, gives 0.

In this project, we aim to derive situations for the existence of a differential equation for when a family of functions do not intersect. We were able to derive a full solution for the result in linear equations, and we offer an exploration of a partial solution for non-linear cases.

Function Families

Fundamentally, function families are functions parameterized by some \(C\), which has the shape:

\begin{equation} y(x, \dots, c) = f(x, \dots)+c \end{equation}

Suppose we analyze first order non-linear system:

\begin{equation} x’ = F(t,x) \end{equation}

We can actually turn this into an autonomous system:

\begin{equation} x_0 = t \end{equation}

\begin{equation} x_0’ = 1 \end{equation}

meaning suddenly we have an autonomous system:

\begin{equation} \begin{cases} x_0’ = 1 \\ x_1’ = F(x_0, x_1) \end{cases} \end{equation}

General strategy:

1. Find zeros of the right side (which are the stationary solutions)
2. Analyze near-stationary solutions through eigenvalues of the linearized Jacobian matrix: if both eigenvalues are zero
3. Away from stationary solutions: basically guessing

Three Examples that are Hopeless to Solve

Lotha-Volterra Prey-Predictor Equation

\begin{equation} \begin{cases} x_1’ = 2x_1-x_1x_2 \\ x_2’ = x_1x_2 - 3x_2 \end{cases} \end{equation}

“Chaotic Dynamics” Because the word is sadly nonlinear.

motivating non-linearity

\begin{equation} \dv t \mqty(x \\ y) = f\qty(\mqty(x\\y)) \end{equation}

This function is a function from \(f: \mathbb{R}^{2}\to \mathbb{R}^{2}\). All the work on Second-Order Linear Differential Equations, has told us that the above system can serve as a “linearization” of a second order differential equation that looks like the follows:

\begin{equation} \dv t \mqty(x \\y) = A \mqty(x \\ y) +b \end{equation}

kernel density estimation

If your data is continuous, you can integrate over the entire dataset and normalize it to be able