Your work should be neat and problems should be completed in the correct order

Your work should be neat and problems should be completed in the correct order. Submit the assignment as a PDF on Blackboard. Write down any intes or antiderivative that you evaluate to solve the equation. Show your sup- porting work if the integral requires simplification, substitution or integration by parts. If you use any method that is not covered in the notes/book, you must explain the method that you used provide a reference in order to receive credit. Problems 1. Consider y' = yes - 9y ey (a) Draw the phase diagram. Identify each equilibrium as stable or unstable. (b) Find lim y(t) for the solution that satisfies the initial condition: y(0) = 1 (c) Is it possible to find a value for yo such that the solution to the IVP: y' = yes - y ey y (0 ) = yo diverges to co as t - co? Why or why not? 2. A second order chemical reaction involves the interaction of one molecule of a substance P with one molecule of a substance Q to produce one molecule of a new substance X. We denote this by: P + Q - X Suppose that p and q, where p # q, are the initial concentrations of P and Q, respectively, and let x(t) be the concentration of X at time t. Then p - x(t) and q - x(t) are the concentrations of P and Q at time t, and the rate at which the reaction occurs is given by the equation: dx dt = a(p - 2)(9-2) where a > 0 is a constant. (a) Assume that 0 < p < q. Draw the phase diagram and label each equilibrium as stable or unstable. (b) If a (0) = 0, determine the limiting value of a(t) as t -+ co. 3. Consider the differential equation: ( et + y ) da + ( 2 + x + ye" ) dy = 0 (a) Verify that the equation is exact. CS Scanned with CamScanner

Autonomous Equal (b) Find a potential function F(x, y). (c) Solve the differential equation. 4. Consider the differential equation: ( y 3 + kay * - 2 ) + ( 3x32 + 202 2 ,3 ) dy = 0 (a) Find the value of k so that the differential equation is exact. (b) Solve the differential equation using your value of k from part (a). CS Scanned with CamScanner