Differential Equations and

Slope Fields6.1

Differential Equations

An equation involving a derivative is called a differential equation.

The order of the differential equation is determined by the highest ordered derivative.

Examples

Determine the order:3 43sin

dyx x x

dx

23 4

23sin

dy d yx x x

dx dx

44

43 2

dy d yy x x

dx dx

42 '( ) 2 ''( ) '''( )x f x y f x f x x

Solving Differential Eq.

3 43sindy

x x xdx

2sec 2 5dy

x xdx

Initial Value

Find the particular solution to the equation whose graph

passes through the point (1,0)

26xdye x

dx

Slope Fields

A diagram depicting the slopes of a differential equation at different points. Gives you an idea of what the graph should look like.

Example

Construct a slope field for Sketch the particular solution

through (0,2).

dyx

dx

Example (implicitly defined)

Construct a slope field for the equation

dy x

dx y

Homework

Pg 327 #12, 15, 18-20, 26-34 even, 58