I have come across the following question when revising for my upcoming exam, and wondered if anyone wouldn't mind giving me a hand and some hints as how to solve it.

And now I'm stuck as to the solution of the equation, as its a 2nd order non-homogenous differential equation, but doesn't have a term in s. Is this a problem when solving this, or do I just put it =0?

Any hints would be appreciated!

Apologies if the symbols don't come out as anticipated. Its my first time using them!

[tex]v(t)=\frac{ds}{dt}\mbox{ so that }\frac{dv}{dt}=\frac{d^2s}{dt^2}[/tex]

and the DE becomes a first order, namely

[tex]F_{m} - kv = m\frac{dv}{dt},[/tex]

that you can solve. Once you have a solution for v, plug it into [tex]v(t)=\frac{ds}{dt},[/tex] and that'll give you the solution to the given DE. Note that we didn't turn a second order DE into a first order DE, but rather a system of 2 first order DEs.

which I then rearranged to make v the subject. However I don't think that this is correct way to do this is it? I think I probably used the wrong technique when integrating?

differs only by a constant term, let us solve for an arbitrary constant as a particular solution, say [tex]v_P(t)=B[/tex], whose derivative is 0, and it must satisfy

[tex]0+\frac{k}{m}B =\frac{f_{m}}{m},[/tex]

so [tex]B =\frac{f_{m}}{k}=v_P(t),[/tex] and hence the most general solution to the given DE (the one in v) is [tex]v(t)=v_H(t)+v_P(t)[/tex], that is