Differential Equation

An equation that involves a derivative

Classifications

Type of Derivative
- Ordinary Differential Equation
- Partial Differential Equation
Order
- Highest order derivative in the equation
Linear or Nonlinear
- $a_{n} (x) \frac{d ^{n} y}{d x ^{n}} + a_{n - 1} (x) \frac{d ^{n - 1} y}{d x ^{n - 1}} + \dots + a_{1} (x) \frac{d y}{d x} + a_{0} (x) y = g (x)$
- The dependent variable $y$ and all its derivatives are of first degree (exponent is 1)
- The coefficients $a_{0}, a_{1}, \dots$ of $y, y', \dots$ and $g (x)$ depend at most on the independent variable $x$
Autonomous
- $\frac{d y}{d x} = f (y)$
- The independent variable is missing form the right side
- Slopes corresponding to two different points with the same y-coordinate must be equal → we can get infinite solutions from one solution by shifting it left or right

\frac{d dependent}{d independent}