Cauchy–Euler equation

It is sometimes referred to as an equidimensional equation.

Because of its particularly simple equidimensional structure, the differential equation can be solved explicitly.

Let y(n)(x) be the nth derivative of the unknown function y(x).

Then a Cauchy–Euler equation of order n has the form

Alternatively, the trial solution

can be used to solve the equation directly, yielding the basic solutions.

[1] The most common Cauchy–Euler equation is the second-order equation, which appears in a number of physics and engineering applications, such as when solving Laplace's equation in polar coordinates.

Substituting into the original equation leads to requiring that

Rearranging and factoring gives the indicial equation

To get to this solution, the method of reduction of order must be applied, after having found one solution y = xm.

This form of the solution is derived by setting x = et and using Euler's formula.

We operate the variable substitution defined by

We analyze the case in which there are distinct roots and the case in which there is a repeated root: If the roots are distinct, the general solution is

If the roots are equal, the general solution is

Observe that we can write the second-order Cauchy-Euler equation in terms of a linear differential operator

We can then use the quadratic formula to factor this operator into linear terms.

denote the (possibly equal) values of

which one can recognize as being amenable to solution via an integrating factor.

and recognizing the left-hand side as the derivative of a product, we then obtain

we substitute the simple solution xm:

For a fixed m > 0, define the sequence fm(n) as

Applying the difference operator to

where the superscript (k) denotes applying the difference operator k times.

Comparing this to the fact that the k-th derivative of xm equals

suggests that we can solve the N-th order difference equation

in a similar manner to the differential equation case.

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 N-th order linear difference equation is also the linear combination of N linearly independent solutions.

Applying reduction of order in case of a multiple root m1 will yield expressions involving a discrete version of ln,

instead (or simply use it in all cases), which coincides with the definition before for integer m.