^ 4. Consider the following autonomous first-order differential equation.= y² (y² −9).dydxFind the critical points and phase portrait of the above differential equation. Classify each criticalpoint as asymptotically stable, unstable, or semi-stable.5. Solve the given differential equation by separation of variables. Solve for YdydxD'e+x+Sy= e

For an autonomous system of the form: , it can be written that:The region above or below the…

Answered: 4. Consider the following autonomous…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

^ 4. Consider the following autonomous first-order differential equation.
= y² (y² −9).
dy
dx
Find the critical points and phase portrait of the above differential equation. Classify each critical
point as asymptotically stable, unstable, or semi-stable.
5. Solve the given differential equation by separation of variables. Solve for Y
dy
dx
D'
e+x+Sy
= e

Expert Solution

Step 1: Explanation

For an autonomous system of the form: $fraction numerator d y over denominator d x end fraction equals f left parenthesis y right parenthesis$ , it can be written that:

The region above or below the equilibrium values on the phase line determines the stability of the critical points.
The critical points are calculated by solving the equation: $f left parenthesis y right parenthesis equals 0$ .
If f(y) is positive, then the region is increasing. If f(y) is negative, then the region is decreasing.
If the region about the equilibrium value is increasing above and decreasing below, then the critical point is unstable.
If it is increasing below and decreasing above, the critical point is asymptotically stable.
If it is increasing below and increasing above or decreasing below and decreasing above, then the critical point is semi-stable.