Oscillation problem. Please help me I'm stupid

[superpig10000]

A particle of mass m is moving under the ocmbined action of the forces -kx, a damping force -2mb (dx/dt) and a driving force Ft. Express the solutions in terms of intial position x(t=0) and the initial velocity of the particle.

For the complementary solution, use x(t) = e^(-bt) A sin (w1t + theta)
And the for the particular solution, use Ct + D

w1^2 = w0^2 - b^2 w0^2 = k/m

Here's what I have so far:
m (dx^2/dt^2) + 2mb (dx/dt) + kx = Ft

(dx^2/dt^2) + 2b (dx/dt) + w0^2 x = At (A = F/m)
The complementary solution is x = e^(-bt) (A1e^(w1t) + A2e^-(w1t). I don't know how to convert this to the form above. And I am totally clueless as to how to find the particular solution.

Please help!

 

[Jhero]

Hello,

Thank you for sharing your work so far. I can definitely help you with finding the particular solution for this problem.

First, let's review the steps for finding the particular solution for a second order differential equation:

1. Determine the form of the particular solution based on the forcing term (Ft in this case). In this case, since the forcing term is a constant (Ft), the particular solution will be in the form of Ct + D.

2. Substitute the particular solution form into the original differential equation and solve for the constants C and D. In this case, we have:

m (d^2x/dt^2) + 2mb (dx/dt) + kx = Ft

Substituting Ct + D for x, we get:

m (d^2/dt^2)(Ct + D) + 2mb (d/dt)(Ct + D) + k(Ct + D) = Ft

Simplifying this, we get:

mC + 2mbC + kC = Ft

Solving for C, we get:

C = Ft / (m + 2mb + k)

Now, let's solve for D. We know that the initial position of the particle (x(t=0)) is given by x(0) = D. Therefore, D = x(0).

3. Putting everything together, the particular solution is:

x(t) = (Ft / (m + 2mb + k))t + x(0)

I hope this helps! Let me know if you have any further questions or need clarification on any of the steps.