I derived a solution for a driven damped oscillator using the DSolve functionality of Wolfram Mathematica. I chose the following Ansatz:

DSolve[{
   0 == -Fmax (1 - Cos[t]) + k x[t] +
   d Derivative[1][x][t] +
   m Derivative[2][x][t],
   x[0] == xStart, x'[0] == vStart}, x[t], t]

Mathematica is able to solve this analytically and I can derive solutions if I pass parameters to the variables.

Anyhow, the derived solution fails at the moment, the given parameters for mass (m), spring constant (k) and dampening (d) fulfill the condition for aperiodic behavior

d=2*Sqrt[m*k]

due to the fact, that a division by zero occurs. The overall solution is given by a big fraction with a denominator containing the following factor

Sqrt[d^2 - 4 k m]

This factor becomes zero in case aperiodic conditions occur which leads to a devision by zero.

Now the fun starts ... I case I pass the aperiodic conditions before giving the system to DSolve:

DSolve[{
    0 == -Fmax (1 - Cos[t]) + k x[t] + 
    2 Sqrt[k m] Derivative[1][x][t] +
    m Derivative[2][x][t],
    x[0] == xStart, x'[0] == vStart}, x[t], t]

DSolve easily derives a working solution without dividing by zero.

It seems to me, as Mathematica is simplifying the more general solution, in a way that I can't use the aperiodic conditions anymore, even though there is a working solution for that case. My question is, can I somehow tell DSolve, to take into account the aperiodic conditions?

Sure, I can workaround the problem with Piecewise[], but for further calculations a general solution without Piecewise[] would be more suitable.

Thanks in advance for you time and help,

Greetings Tschibi