Solving the Differential equation x″=−kx+c Usually the best method for solving the ODE for the Differential equation of an SHM would be to take the trial solution $e^{mt}$ and hence solving the auxiliary equation. But clearly the 'c' term will pose a problem for these kinds of equations. So how do I solve them?

$$x''+kx=c$$
Just rewrite the DE as:
$$\left(x- \dfrac c k\right)''+k\left(x-\dfrac ck\right)=0$$
$$y''+ky=0$$
Where $y=x-\dfrac  ck$
For $k=0$ just integrate the DE.
