3. Direction Fields

Direction Field

Definition

Given a first order differential equation

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

a direction field is given by drawing a small line piece with slope $y^{\prime }$ satisfying $F(a,b,y^{\prime })=0$ at many points $(a,b)$ in the plane.

Example

$T^{\prime }=-(T-5)$

where $T(0)=10$ and equalling

$T(t)=5+5e^{-t}$

Autonomous Equation

Definition

A differential equation $F(x,y,y^{\prime },\dots ,y^{(n)})=0$ is called autonomous if it does not depend explicitly on the independent variable, i.e.

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

Equilibrium Solution

Definition

An equilibrium solution of a differential equation

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

is a solution $y(x)$ such that $y^{\prime }(x)=0$.

Example

The cooling equation $T^{\prime }=-(T-5)$ has equilibrium solution $T(t)=5$ which can be found by solving the equation $0=-(T-5)$

obtained by setting $T^{\prime }=0$.