### Solution Found!

# Consider the three systems (i) dx dt = 2x + y dy dt = y +

**Chapter , Problem 1**

(choose chapter or problem)

**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.]

### 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 and to obtain their value at :

Therefore, the Jacobian matrix at the equilibrium point

=

in this case is given by:

=