Consider the differential equation d x(t) = 4x − y dt d dt y(t) = 2x + y. Write the differential equation in matrix form. Characterise the equilibrium at the origin and solve the initial value problem where x(0) = 3, y(0) = 4.

Consider the following system of differential equations describing two populations: æ' = 0.02a (1 – ) + 0.004xy 80 (1 y' = 0.04y(1 ) - 0.0002xy 20 a. Describe what the equations tell us about the populations (how the populations behave absent interaction, and how interaction affects each population). b. Find all equilibrium points algebraically. c. Classify each equilibrium using linearization d. Sketch some solution curves in the phase plane (x-y plane)

Can you calculate X(t) = exp(At) and show the matrix satisfies the differential equations. And please answer the final part - characterise the trivial equilibrium and sketch the phase portrait in the new coordinates.

Determine the values of r for which the given differential equation has solutions of the form y = e't. Give the answers in ascending order. y"-6y" + 8y = 0 r = r = r = Jud L IN