(a) Two solutes X and Y are mixed in a beaker. Their respective concentrations x(t) and y(t) satisfy the following differential equations:

dx dt = x − xy − μx2, dy dt = −y + xy − μy2.

Find and classify the critical points forμ > 0, and plot possible phase portraits showing the different types of qualitative behavior. Interpret the results in terms of the concentrations of solutes X and Y .

[14]

(b) Determine the Hamiltonian of the system dx dt = y, dy dt = x − x2.

Plot a phase portrait.

[6]