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Consider the two differential equations da dt = (ra)(x - b)(x-c), and (a-x)(b-x)(c-x), dt each having the critical points a, b and c; suppose that a < b < c. For one of these equations, only the critical point b is stable; for the other equation, b is the only unstable critical point. Construct phase diagrams for the two equations to determine which is which. Without attempting to solve either equation explicitly, make rough sketches of typical solution curves for each.

Consider the two differential equations da dt = (ra)(x - b)(x-c), and (a-x)(b-x)(c-x), dt each having the critical points a, b and c; suppose that a < b < c. For one of these equations, only the critical point b is stable; for the other equation, b is the only unstable critical point. Construct phase diagrams for the two equations to determine which is which. Without attempting to solve either equation explicitly, make rough sketches of typical solution curves for each.

Advanced Engineering Mathematics
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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(6) Consider the two differential equations da dt (ra)(x - b)(x-c), and (a-x)(b-x)(c-x), each having the critical points a, b and c; suppose that a <b< c. For one of these equations, only the critical point b is stable; for the other equation, b is the only unstable critical point. Construct phase diagrams for the two equations to determine which is which. Without attempting to solve either equation explicitly, make rough sketches of typical solution curves for each.
Transcribed Image Text:(6) Consider the two differential equations da dt (ra)(x - b)(x-c), and (a-x)(b-x)(c-x), each having the critical points a, b and c; suppose that a <b< c. For one of these equations, only the critical point b is stable; for the other equation, b is the only unstable critical point. Construct phase diagrams for the two equations to determine which is which. Without attempting to solve either equation explicitly, make rough sketches of typical solution curves for each.
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