Math 240 Midterm Please show all your work. 1. A ball is thrown upward from the top of a 200 foot tall building with an initial velocity of 40 feet per second. Using \"up\" as the positive direction and ground level as the origin, write the initial value problem for the height x of the ball at time t. dy 2. A DE in the form dx + P (x)y = 0 is both linear and separable. Show that it can be solved by both techniques, separation and an integrating factor, and that the result will be the same either way. = P (8 2P ) with initial condition P (0) = 2 and 3. Solve the logistic equation dP dt identify the critical values for the DE as stable, unstable or semi-stable. R R 4. Show why we can use P (x) dx instead of P (x) dx + C when finding an integrating factor for a first order linear DE. 5. Without solving for the undetermined coefficients, what is the correct form of the particular solution of the differential equation y 00 4y 0 + 5y = e2x sin(x)? 6. Solve the differential equation y 00 3y 0 = 6x 2. 7. Solve the differential equation y 00 + 9y = sin(3x). 8. Solve the differential equation x3 y 000 3x2 y 00 + 6xy 0 6y = 0. 9. Find the first four nonzero terms in the Taylor expansion about x = 0 of the solution of the initial value problem y 00 = ey , y(0) = 1, y 0 (0) = 1. 10. Find the solution to the system of differential equations dx = 5x 4y dt dy =x dt making sure to relate the constants of the solutions for x and y