differential equations and slope fields 6.1. differential equations an equation involving a...
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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
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