How do I solve this forced oscillation differential equation?

[Lizwi]

Homework Statement

Please help me solve this differential equation: This is mass attached to a spring, we have

F= ma= -kx -bv + F_ext where k and v are spring constant and velocity respectively and F_ext is an additional external force.

Homework Equations

I know how to solve nonhomogeneous differential equations mathematically but on the right of the above equation is not a function so I stuck.

The Attempt at a Solution

I tried this way:
writing the above as differential equation I have
md²x/dt² + bdx/dt + kx = F_ext
for homogeneous part md²x/dt² + bdx/dt + kx=0
the solution I assumed is x(t) =Ae^pt
the first derivative of the assumed solution is x'(t)= Ape^pt and the second derivative is x''(t)= Ap²e^pt
substituting all these x(t), x'(t) and x''(t) to the differential equation and divide by
Ae^pt I get:

mp² + bp+k=0
for inhomogeneous part I don't know ho to handle this right hand side F_ext. Please help me.

 

[tiny-tim]

Hi Lizwi!

… and F_ext is an additional external force.

Is F_ext a constant?

You need to find any solution for the whole equation …

try polynomials first (starting with constants!) ​

 

[rude man]

You need to know the explicit expression for your driving function first, so you can "guess" at the inhomogeneous solution. What is it?

 

[OldEngr63]

You could express the solution for any right side function by means of a convolution integral, but that is only a symbolic solution and not good for too much.

 

[sgd37]

just absorb the external force into x i.e. x' = x + F. If it's a constant then it's not too difficult to solve, if it isn't then you have another (manageable) diff equation to solve.