Sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.-6-4-2-2y2246X-6-4-2-22246X Consider the following autonomous first-order differential equation.dy= y²(4- y²)Find the critical points and phase portrait of the given differential equation.unstable2semi-stable040-402Classify each critical point as asymptotically stable, unstable, or semi-stable. (List the critical points according to their stability. Enter your answers as a comma-separated list. If there are no critical points in a certain category, enter NONE.)asymptotically stable0

Given differential equation is dydx=y24-y2For critical points we have dydx=0⟹…

Answered: ketch typical solution curves in the…

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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Related questions

Question

Sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.
-6
-4
-2
-2
y
2
2
4
6
X
-6
-4
-2
-2
2
2
4
6
X

Consider the following autonomous first-order differential equation.
dy
= y²(4- y²)
Find the critical points and phase portrait of the given differential equation.
unstable
2
semi-stable
0
4
0
-4
0
2
Classify each critical point as asymptotically stable, unstable, or semi-stable. (List the critical points according to their stability. Enter your answers as a comma-separated list. If there are no critical points in a certain category, enter NONE.)
asymptotically stable
0

Expert Solution

Step 1

$\begin{array}{l} G i v e n d i f f e r e n t i a l e q u a t i o n i s \\ \frac{d y}{d x} = y^{2} (4 - y^{2}) \\ F o r c r i t i c a l p o i n t s w e h a v e \frac{d y}{d x} = 0 \\ ⟹ y^{2} (4 - y^{2}) = 0 \\ y^{2} (2 - y) (2 + y) = 0 \\ y = 0, 0, 2, - 2 \\ W e h a v e c r i t i c a l p o i n t s 0, 0, 2, - 2 . \\ N o w w e h a v e \\ \frac{d y}{d x} < 0 f o r y < - 2 ⟹ y (x) d e c r e a s i n g \\ \frac{d y}{d x} ⩾ 0 f o r - 2 ⩽ y ⩽ 2 ⟹ y (x) i n c r e a s i n g \\ \frac{d y}{d x} < 0 f o r y > 2 ⟹ y (x) d e c r e a s i n g \end{array}$