Question 1 (30 marks) (1) Solve the nonlinear differential equation dy - 3y - 1' = 2, #(0) = 0. (2) Determine the maximum time for such that a solution may exist. 3) Without solving the equation, determine the highest lower bound bg on the time such that a solution may exist. Question 2 (15 marks) Find the second in the sequence of successive approximations to the solution of dy dt + 1+ 2 8 +Ey = 0, #(0 ) = 1. Question 3 (15 marks) Determine the equilibrium points for the differential equation dt * - 20y' + 64. Draw a direct-line vector field. Determine which equilibrium points are stable and which are not. Question 4 (40 marks) In the following circuit, the current source is a constant DC current Io. The voltage Vo on the capacitor and the current through the inductor Ir are all initially equal to zero, i.e., Vc(0) = IL(0) = 0. Current BOUTCO Resistor Inductor Capacitor E (1) Derive a differential equation for Vo(t). (2) Let R = 2, L = 5/3, and C = 1/10. Solve for Vo(t). (3) If we replace the constant ly by Jo(t) = et but keep the same initial conditions, what will be Volo)