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Math / Differential Equations 00 4 / Chapter 5.1 / Problem 3

Differential Equations 00 | 4th Edition | ISBN: 9780495561989 | Authors: Paul (Paul Blanchard) Blanchard, Robert L. Devaney, Glen R. Hall

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Consider the system dx dt = 2x + y dy dt = y + x2. (a)

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QUESTION:

Consider the system dx dt = 2x + y dy dt = y + x2. (a) Find the linearized system for the equilibrium point (0, 0). (b) Classify (0, 0) (as either a source, sink, center, . . . ). (c) Sketch the phase portrait for the linearized system near (0, 0). (d) Repeat parts (a)(c) for the equilibrium point at (2, 4).

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QUESTION:

Consider the system dx dt = 2x + y dy dt = y + x2. (a) Find the linearized system for the equilibrium point (0, 0). (b) Classify (0, 0) (as either a source, sink, center, . . . ). (c) Sketch the phase portrait for the linearized system near (0, 0). (d) Repeat parts (a)(c) for the equilibrium point at (2, 4).

ANSWER:

Step 1 of 7

a) To solve part a of this question, we use the jacobian to study the behavior of the system around

=

=

=

By calculating the Trace and Determinant of this matrix and comparing it to the Trace-Determinant Plane, we can make a conclusion about the behavior of the system near .

Since the trace is negative and the determinant is positive and the determinant is less than , we can conclude that the system is global sink near

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