I'm trying to calculate the position of a particle in a quadrapole magnet depending on the entry position $x_0$ and the combined (constant) physical parameters $k$. Given an equation
$$x(t) =\frac{(\frac{x''(t)}{k})''}{k},$$
solving via assuming that $x(t) = e^{\lambda t}$ et,c...
I arrive at the general solution
$$x(t) = c_1\cos(\sqrt{k}\cdot t)+c_2\sin(\sqrt{k}\cdot t)+c_3e^{-\sqrt{k}\cdot t}+c_4e^{\sqrt{k}\cdot t}$$
with $c_1,c_2,c_3,c_4$ arbitrary constants. What would it mean mathematically if I were to set say $c_1,c_2,c_3 = 0$, assuming I don't have other constraints (in my example I would have additional $x(0) = x_0$, but as far as I can see that doesn't forbid it).
Given that they are arbitrary, I can't see a problem with it. Of course, if you have additional starting conditions, you have to set the constants accordingly, but in my example $c_4 = x_0$ seems to do the job and leaves me with a much simpler solution. So why would I ever NOT eliminate every unnecessary term?