Let $a,b,c$ be real constants with $a>0$, and let the differential equation $a{{d^{2}y}\over{dx^{2}}}+b{{dy}\over{dx}}+cy=0$ have auxiliary equation $as^{2}+bs+c=0.$

(i) If there are distinct real roots $p$ and $q$, then the general solution is

(ii) if there is a double real root $p$, then the general solution is

(iii) if there is a pair of complex conjugate roots $\alpha\pm i\beta$, then