Solution Found!
Consider the three systems (i) dx dt = 2x + y dy dt = y +
Chapter , Problem 1(choose chapter or problem)
Consider the three systems (i) dx dt = 2x + y dy dt = y + x2 (ii) dx dt = 2x + y dy dt = y + x2 (iii) dx dt = 2x + y dy dt = y x2. 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 three systems (i) dx dt = 2x + y dy dt = y + x2 (ii) dx dt = 2x + y dy dt = y + x2 (iii) dx dt = 2x + y dy dt = y x2. 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
in this case is given by:
