Reference no: EM131064865

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.