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MATH2023 - Study Period 5 2021

Practice Question

A. Consider the following ODE

Questions to answer during zoom:

(i) What can you tell me about this ODE? What kind of equation is it?

(ii) What technique might you use to solve it?

(iii) Are there any potential problems to be aware of when using this technique?

Question to solve in your handwritten work after zoom:

(iv) Solve this differential equation for .

(2+1+2+4=9 marks)

B. Consider the following ODE

This equation has as a particular solution.

Questions to answer during zoom:

(i) What can you tell me about this ODE? What type of equation is it?

(ii) What does it mean that is a particular solution?

(iii) What technique might you use to solve it? (You do not need to solve this equation during the writing time.)

(2+1+1=4 marks)

C. You used Matlab to produce a phase portrait of the following system of equations

Unfortunately you forgot to label the figure and it got mixed up with another one.

The two phase portraits in question are shown below.

Your notes tell you that your system of equations has equilibrium points at (0, 0) and (2, 2), and that the Jacobian of your system is given by

Questions to answer during zoom:

(i) How can you use the Jacobian to classify the critical points in terms of their type and stability?

(ii) For the critical point at (0, 0), your notes say What is ? What does this tell you about the point (0, 0)?

(iii) For the critical point at (2, 2), your notes say What does this tell you about the point (2, 2)? What might you say if you had similar information from a linear system of ODEs?