Solution Found!
Consider the following three systems: (i) dx dt = 3 sin x
Chapter , Problem 2(choose chapter or problem)
Consider the following three systems: (i) dx dt = 3 sin x + y dy dt = 4x + cos y 1 (ii) dx dt = 3 sin x + y dy dt = 4x + cos y 1 (iii) dx dt = 3 sin x + y dy dt = 4x + 3 cos y 3. All three have an equilibrium point at (0, 0). Which two systems have phase portraits with the same local picture near (0, 0)? Justify your answer. [Hint: Very little computation is required for this exercise, but be sure to give a complete justifi- cation.]
Questions & Answers
QUESTION:
Consider the following three systems: (i) dx dt = 3 sin x + y dy dt = 4x + cos y 1 (ii) dx dt = 3 sin x + y dy dt = 4x + cos y 1 (iii) dx dt = 3 sin x + y dy dt = 4x + 3 cos y 3. All three have an equilibrium point at (0, 0). Which two systems have phase portraits with the same local picture near (0, 0)? Justify your answer. [Hint: Very little computation is required for this exercise, but be sure to give a complete justifi- cation.]
ANSWER:Step 1 of 7
As stated, all three given systems have an equilibrium point at 
For the first system, let:
Then, their partial derivatives are given by:
Substitute 0 for both 
Therefore. the Jacobian matrix at. the equilibrium point
is given by:
The linearized system
in this case is given by:
