Question
Asked Jul 30, 2019
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How does the limiting behavior of solutions depend on the value of y(0)?

18. Make a rough sketch of a direction field for the autono
mous differential equation y' fy), where the graph of
f is as shown. How does the limiting behavior of solutions
depend on the value of y(0)?
f(y)A
2
-1 0
-2
1
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18. Make a rough sketch of a direction field for the autono mous differential equation y' fy), where the graph of f is as shown. How does the limiting behavior of solutions depend on the value of y(0)? f(y)A 2 -1 0 -2 1

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Step 1

Given autonomus differential equation:

Graph of function f is given.
We need to show the the limiting behavior of solution depend on the value of y(0)

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у'- f). f(y)A -2 1 0 2 у т чфов

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Step 2

Since given graph is symmetric about the vertical y -axis. So, f(-y) = f(y). It means slope is not change the reflection about the horizontal axis. Point (x,y) and (x, -y) are same.

Since value of function is 0 :

the slopes are horizontal on these four lines, giving rise to four constant solutions.

Given graph\'s slope doesn\'t depend on x, it means direction field doesn\'t change under the horizontal shift.

 

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fy) 0;y 12 Direction Field

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Step 3

Given graph is positive for interval:

In these regions direction fi...

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yE (-2)(1,1)(2,)

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Tagged in

Math

Calculus

Differential Equations

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