Design a script file which solves each of the following problems in a separate cell. Please take the time to answer any questions that may be asked in the form of a comment in the appropriate cell. 1. Consider the differential equation y-y-2cost, y(0) = a a. Draw a direction field. How do solutions appear to behave as t. Does the behavior seem to depend on the choice of a? Let a, denote the value for which the transition from one type of behavior to another occurs. Estimate a, from your direction field. b. Solve the initial value problem and find the critical value a, exactly. c. Describe the behavior of the solution corresponding to the initial value a = a. 2. Consider the initial value problem y+y=3+2cos 20 -2 cos 21, y(0)-0 a. Find the solution of this initial value problem and describe its behavoir for large t. b. Determine the value of t for which the solution first intersects the line y = 12. 3. Solve the initial value problem 2 cos2x 3+2y y(0)=-1 and determine where the solution attains its maximum value. 4. Draw a direction field and plot several solutions of the differential equation y'=ty(3-y) Describe how solutions appear to behave as t increases and how their behavior depends on the initial value y, when :-0