The differential equation representing family of curves y = ae^mx + be^nx, where a and b are arbitrary constants, is

\(\frac{d^2y}{dx^2}\)-(m + n)\(\frac{dy}{dx}\) + mny =0

y = ae^mx + be^nx

on differentiating both side, we get

$\frac{dy}{dx}=ame^{mx}+bne^{nx}$

Again differentiating, we get

$\frac{d^2y}{dx^2}=am^2e^{mx}+bn^2e^{nx}$

$∴\frac{d^2y}{dx^2}-(m+n)\frac{dy}{dx}+mny=am^2e^{mx}+bn^2e^{nx}$

$-(m+n)[ame^{mx}+bne^{nx}]+mn[ae^{mx}+be^{nx}]$

$=0$