DEPARTMENT OF MATHEMATICS UNIVERSITY OF KANSAS MATH 220 - FALL 2009 - FINAL EXAM Your Name: On this exam, you may use a calculator and formula notes. It is not sufficient to just write down the answers. You must explain how you arrived at your answers and how you know they are correct. 1 (35) 2 (35) 3 (35) 4 (35) 5 (35) 6 (35) 7 (35) 8 (35) 9 (35) 10 (35) Total (350) 2 • 1. (35 points) Solve the initial-value problem and sketch the graph of the solution 2t y0 + 2 y = 2t(t2 + 1), y(1) = 2. t +1 • 2. (35 points) Find the solution in explicit form and determine the interval of existence of the solution for the initial-value problem t + ye−t y 0 = 0, y(0) = 1. 3 • 3. (35 points) Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 200 liters of dye solution with a concentration of 1g/liter. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 2 liters/min, the well-stirred solution flowing out at the same rate. Find the time that will ellapse before the concentration of dye in the tank reaches 1 % of its original value. 4 • 4. (35 points) Find the solution of the initial-value problem y 00 + 2y 0 + 5y = 0 y(0) = 1 0 y (0) = 0 • 5. (35 points) Find the general solutions of y 00 + 4y = t2 + 3et 5 • 6. (35 points) Solve the exact equation (9x2 + y − 1)dx − (4y − x)dy = 0, y(1) = 0 and determine where the solution is valid. • 7. (35 points) Find the general solution of the linear system 0 x = x+y y 0 = 4x + y. 6 • 8. (35 points) Find the solution of the initial-value problem 0 x1 = x1 − 4x2 x02 = 4x1 − 7x2 . where 3 x(0) = . 2 Draw the graph of the solution and describe its behavior for increasing t. 7 • 9. (35 points) Use Laplace transform to solve the initial-value problem y 00 + 2y 0 + 5y = sin 2t y(0) = 2 0 y (0) = −1 • 10. (35 points) Solve the system of equations 0 x1 = 2x1 − 5x2 x02 = x1 − 2x2