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?
Solving the Differential equation x″=−kx+c
ordinary differential equations
Best Answer
$$x''+kx=c$$ 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.