the coefficient of is zero. Solving the quadratic equation, we get . The general solution is therefore Difference equation analogue There is a difference equation analogue to the Cauchy–Euler equation. For a fixed , define the sequence as Applying the difference operator to , we find that If we do this times, we find that where the superscript denotes applying the difference operator times. Comparing this to the fact that the -th derivative of equals suggests that we can solve the N-th order difference equation in a similar manner to the differential equation case. Indeed, substituting the trial solution brings us to the same situation as the differential equation case, One may now proceed as in the differential equation case, since the general solution of an -th order linear difference equation is also the linear combination of linearly independent solutions. Applying reduction of order in case of a multiple root will yield expressions involving a discrete version of , (Compare with: ) In cases where fractions become involved, one may use instead (or simply use it in all cases), which coincides with the definition before for integer . See also Hypergeometric differential equation Cauchy–Euler operator References Bibliography Ordinary differential equations