Differential Equations and Slope Fields 6.1. Differential Equations An equation involving a...
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![Page 1: Differential Equations and Slope Fields 6.1. Differential Equations An equation involving a derivative is called a differential equation. The order.](https://reader036.fdocuments.net/reader036/viewer/2022083008/56649f3b5503460f94c58ec6/html5/thumbnails/1.jpg)
Differential Equations and
Slope Fields6.1
![Page 2: Differential Equations and Slope Fields 6.1. Differential Equations An equation involving a derivative is called a differential equation. The order.](https://reader036.fdocuments.net/reader036/viewer/2022083008/56649f3b5503460f94c58ec6/html5/thumbnails/2.jpg)
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.
![Page 3: Differential Equations and Slope Fields 6.1. Differential Equations An equation involving a derivative is called a differential equation. The order.](https://reader036.fdocuments.net/reader036/viewer/2022083008/56649f3b5503460f94c58ec6/html5/thumbnails/3.jpg)
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
![Page 4: Differential Equations and Slope Fields 6.1. Differential Equations An equation involving a derivative is called a differential equation. The order.](https://reader036.fdocuments.net/reader036/viewer/2022083008/56649f3b5503460f94c58ec6/html5/thumbnails/4.jpg)
Solving Differential Eq.
3 43sindy
x x xdx
2sec 2 5dy
x xdx
![Page 5: Differential Equations and Slope Fields 6.1. Differential Equations An equation involving a derivative is called a differential equation. The order.](https://reader036.fdocuments.net/reader036/viewer/2022083008/56649f3b5503460f94c58ec6/html5/thumbnails/5.jpg)
Initial Value
Find the particular solution to the equation whose graph
passes through the point (1,0)
26xdye x
dx
![Page 6: Differential Equations and Slope Fields 6.1. Differential Equations An equation involving a derivative is called a differential equation. The order.](https://reader036.fdocuments.net/reader036/viewer/2022083008/56649f3b5503460f94c58ec6/html5/thumbnails/6.jpg)
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.
![Page 7: Differential Equations and Slope Fields 6.1. Differential Equations An equation involving a derivative is called a differential equation. The order.](https://reader036.fdocuments.net/reader036/viewer/2022083008/56649f3b5503460f94c58ec6/html5/thumbnails/7.jpg)
Example
Construct a slope field for Sketch the particular solution
through (0,2).
dyx
dx
![Page 8: Differential Equations and Slope Fields 6.1. Differential Equations An equation involving a derivative is called a differential equation. The order.](https://reader036.fdocuments.net/reader036/viewer/2022083008/56649f3b5503460f94c58ec6/html5/thumbnails/8.jpg)
Example (implicitly defined)
Construct a slope field for the equation
dy x
dx y
![Page 9: Differential Equations and Slope Fields 6.1. Differential Equations An equation involving a derivative is called a differential equation. The order.](https://reader036.fdocuments.net/reader036/viewer/2022083008/56649f3b5503460f94c58ec6/html5/thumbnails/9.jpg)
Homework
Pg 327 #12, 15, 18-20, 26-34 even, 58