Introduction to Differential Equations

Definition

A differential equation (DE) is any equation in which the unknown is a function.

A DE always includes derivatives of the function.

The order of a DE is the highest derivative that appears.

Will only go over one independent variable DE’s in this course, typically with variable $t$ (time).

Example

$$\frac{dy}{dt}=y^2+t^2$$

This DE has order 1, or it is of the first-order.

That is the same with $y^{\prime }=y$.

Example

$$\frac{d^{2}y}{d{t}^2}+y\frac{dy}{dt}+(\frac{dy}{dt}{)}^{10}=0$$

The first section ($\frac{d^{2}y}{d{t}^2}$) has the highest order, which is 2, so this DE has an order of 2.

Ordinary Differential Equations