ODEs are Differential Equations in one independent variable: \(y(x)\).

Main Content:

- First-Order Differential Equations
- Second-Order Linear Differential Equations
- Uniqueness and Existance

## Overarching Categories

### order of equations

the order of an equation is the highest derivative of an equation

### linear vs. non-linear differential equations

A solution of a differential equation is **linear** when solutions are closed under linear operations.

We can spot an ODE by seeing that each of its derivatives are seperated or in separable terms, and only to the first power—because that ends up being a linear equation (i.e. any two solutions satisfying the equation can add and scale to another solution).

The RHS doesn’t matter. For instance:

\begin{equation} xy’’ + e^{x}y’ + (x^{2}-3)y = x^{2}-x \end{equation}

is linear.

#### superposition principle

any linear combination of a *homogeneous linear* ODE is also a solution to the ODE.

functional linear independence

Recall linear independence. If we have two solutions \(y_1\), \(y_2\), are linearly independent or “independent”, if

\begin{equation} c_1 y_1(t) + c_2y_2(t) = 0 \end{equation}

implies \(c_1 = c_2 = 0\).

### homogeneous vs. inhomogeneous equations

whether or not, isolating all the DEPENDENT variables to the left side, is the right side zero?

### linear systems

systems of ODEs are groups of ODEs. Linear systems can obtain you a vector-value function:

\begin{equation} y’(x) = \mqty(3 & -2 \\ -1 & 5) \vec{y} \end{equation}