d^2/dx^2(q(x))-5*(d/dx(q(x)))+6*q(x)=0
This is a homogeneous second order differential equation with constant coefficients.
q"-5q'+6=0 ... (1)
We solve the auxiliary equation:
m²-5m+6=0
(m-2)(m-3)=0
m=2 or 3
The solution to the homogeneous differential equation is thus:
q= C1*e^(2) + C2*e^3