Math 232, Test 2, 14 February 2008Name:Instructions.Do each of the following questions. Please show all of your work. Good Luck!1.(Short Answers) (a) SupposedYdt=AYis a system of differential equations whereAhas eigenvalues with corresponding eigenvectors: 2,12and-3,1-1. Write the generalsolution to this system.(b) Write the second-order differential equationd2ydt2-5dydt+ 6y=etas a system of first-orderdifferential equations.(c) Determine whetherY(t) = (e-t,3e-t) is a solution to the systemdxdt= 2x-ydydt=x-2y.(d) Is-21an eigenvector for the matrix2213? If so, find its corresponding eigenvalue.(e) Consider the systemdxdt= 2x1-x2-xyanddydt= 3y1-y3-2xy. Is it a predator-prey system, or a competing species system? Explain.
2.Find all equilibrium points for the systemdxdt=x(2-x-y)dydt=y(y-x)3.Find the general solution to the partially decoupled system