Cauchy–Euler equation

The Euler differential equation (after Leonhard Euler ) is a linear ordinary differential equation of higher order with non-constant coefficients of the special form

Given to and inhomogeneity. If we know a fundamental system of the homogeneous solution, so you can determine the general solution of the inhomogeneous equation by the method of variation of constants. Therefore, only needs to be considered.

Euler's equation is converted by means of transformation to a linear differential equation with constant coefficients.

Motivation of the transformation

Be a sufficiently smooth function and

Then we have

So

In this respect, the Euler second order differential equation into a linear differential equation with constant coefficients would transform. There are now faced with the following questions:

• Convicted this transformation also the higher order terms in which with constant coefficients?
• How can you calculate the coefficients on the right side easier than every time the transformation enough to derive often?

These issues will be clarified by the following transform set:

The transform set

Be solution of linear differential equation with constant coefficients

Then

A solution of the ( homogeneous ) Euler's differential equation

Explanation of Notation

Here, first the differential operators are together (similar to the multiplying out ) links before they are applied to a function, for example:

Evidence

To show is only for all. This is done by induction. The induction base is trivial. Assuming the validity of the identity of this identity can be differentiated. The result is

Applying the induction hypothesis implies

Conclusion: Construction of a fundamental system

The characteristic equation for the differential equation is

Now denote the zeros of the characteristic polynomial and the multiplicity of such forms

A fundamental system of equation. So is

A fundamental system of ( homogeneous ) Euler's differential equation.

Example

Consider the Euler differential equation

To solve the above set initially by the following linear differential equation with constant coefficients

So

The photo associated with this differential equation characteristic polynomial is

And has the zeros

Case 1: Both real.

Then a fundamental system for the transformed linear differential equation. The inverse transformation provides that a fundamental system for the original Euler's differential equation.

Case 2:.

Then is a double root of the characteristic polynomial. Therefore, a fundamental system for the transformed linear differential equation. The inverse transformation provides that a fundamental system for the original Euler's differential equation.

Case 3: both not real.

Then are complex conjugate. So a ( complex ) is fundamental system. Be. Then is a real fundamental system of the transformed linear differential equation. Inverse transformation provides a fundamental system for the original Euler's differential equation.