Note: this exam will be 240 minutes long, closed book, closed notes. You will be allowed to use one 3 5 notecard. Indicate your answer clearly and show all your work. Good luck: show that the following system has two limit cycles (i. e. periodic solutions). Determine the stability of each. x = y + x y = x + y . 1: find the general solution to the di erential equation y + 4y = 3te2t. 2: given that y1(t) = t and y2(t) = tet are solutions of the corresponding homogeneous di erential equation, nd a particular solution using variation of parameters t2 y t(t + 2)y + (t + 2)y = 2t3. 5: assume the coe cients are chosen such that the system is expressed as: x = x(1 x z) y = y(2 x y z) z = z( 1 + 2x) Find the equilibrium points. (hint: there should be 5 points!)
