FinalThis is a closed book exam.You have two hours. Solve the following exercises and write clear solutionsto receive partial credit. Good Luck.1. (a) Find the general solution of the differential equation:y00+ y = 01(b) Solve the initial value problemy00+ y = 6 sin xy(0) = 1y0(0) = 022. Find the general solution of the differential equation y00+ 2y0+ y = 0.33. Consider the differential equationdydx=√x − y. Discuss the existence and uniqueness of thisdifferential equation in terms of the theorem that we discussed in class, with the initial conditiony(2) = 2.44. Find the general solution to the differential equationxeydy + (ey+ x3e2x)dx = 055. Suppose the population P (t) of a country satisfies the differential equationdPdt= kP (200 − P )with k a constant (with P given in millions). Its population in 1940 was 100 million and wasthen growing at a rate of 1 million per year. Predict this country’s population in the year 2010.66. Find the solution to the following initial value problem using Laplace transforms.x00− x0− 2x = 1x(0) = 5x0(0) = 077. Consider the sistem of differential equations:(dxdt= 3x + ydydt= x + y.(a) Write the system in matrix form.(b) Find the eigenvalues.(c) Give the general solution of the system.(d) Solve the initial condition problem with x(0) = 0 and y(0) =