1. Test for exactness. If exact, solve. If not, use an integrating factor (find it by inspection). Also, determine the corresponding particular solution for the given initial condition. e2x(2cos(y)dx - sin(y)dy) = 0, y(0) = 0. 2. Solve the ODE by integration dy/dx + x7e-(x8/8) = 0. 3. State the order of the ODE. Verify that the given function is a solution. (c is an arbitrary constant) y' = 2 + 2y2, y = tan(2x + c). 4. Find the particular solution. e5xy' = 5(x + 4) y6, y(0) = 1/5√21. 5. Reduce to first order and solve. y'' + y'3 sin(y) = 0. 6. Functions e-0.6x and xe-0.6x are linearly independent and form a basis of solutions of the following ODE. Solve the IVP. y'' +1.2y' + 0.36y = 0, y(0) = 2.2, y'(0) = 0.12. 7. Solve the initial value problem. Check that your answer satisfies the ODE as well as the initial conditions. y'' + 4y' - 21y = 0, y(0) = 10, y'(0) = -40. 8. Factor and solve. (D2 + 8.9D + 7.9I) y = 0. 9. Find a real general solution. xy'' + 4y' = 0. 10. Find the general solution of y'' - 81y = 64.8e9x + 320ex. 11. Find the steady state oscillation of the mass spring system modeled by the given ODE. y'' + 8y' + 15y = 717.5cos(4t). 12. Solve the given non-homogeneous ODE by variation of parameters or undetermined coefficients. Give a general solution. (D2 - 4D + 4I)y = x-10e2x 13. Solve the initial value problem. y''' + 3.2y'' + 4.81y' = 0, y(0) = 4.1, y'(0) = -6.10, y''(0) = 14.71. 14. Solve the initial value problem. (D3 + 4D2 + 85D)y = 135xex y(0) = 13.9, Dy(0) = -18.1, D2y(0) = -691.6. 15. Find a real general solution of the following system y1' = 3y1 - y2, y2' = 9y1 + 9y2.