2. Introduction to Differential Equations

Differential Equations

Definition

A differential equation is an equation of the form:

$F(x,y,y^{\prime },y^n,\dots ,y^{(n)})=0$

where $y=y(x)$ is a function of the variable $x$.

• We call $n$ the order of the differential equation.
• The variable $x$ is called the independent variable
• and the variable $y$ is called the dependent variable.

General Solution

Definition

Given a differential equation

$F(x,y,y^{\prime },y^n,\dots ,y^{(n)})=0$

a general solution is given by a formula for a function $y(x)$ that depends on $n$ independent arbitrary constants $c_1,\dots ,c_n$ such that any particular solution is obtained by choosing the appropriate values for the constants.

Initial Value Problem

Definition

An nth order initial value problem is given by a differential equation

$F(x,y,y^{\prime },\dots ,y^{(n)})=0$

together with prescribed values at some initial point $x=a$:

$y(a)=y_0$

$y^{\prime }(a)=y_1$

$\dots$

$y^{(n)}(a)=y_n$