Find the steady state oscillation of the mass spring system

Question

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.

e^2x(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.

e^5xy' = 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 = 64.8e^9x + 320e^x.

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^-10e^2x

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 = 135xe^x

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.