The Euler equation (by Leonhard Euler ) is a linear differential equation of higher order with non-constant coefficients of the particular form
to given and inhomogeneity . If one knows a fundamental system of the homogeneous solution, one can determine the general solution of the inhomogeneous equation with the procedure of the variation of the constants . Therefore only needs to be considered.
Euler's differential equation is converted into a linear differential equation with constant coefficients by means of the transformation .
Motivation of the transformation
Be a sufficiently smooth function and
-
, so .
Then applies
so
In this respect, Euler's second order differential equation would transform into a linear differential equation with constant coefficients. The following questions now arise:
- Does this transformation also convert the higher order terms into those with constant coefficients?
- How can one calculate the coefficients on the right-hand side more easily without deriving the transformation enough times each time?
These questions are clarified by the following transformation theorem:
The transformation set
Let the linear differential equation be solved with constant coefficients
Then
a solution of Euler's (homogeneous) differential equation
Explanation of the notation
Here, the differential operators are first linked with each other (comparable to multiplying) before they are applied to a function, for example:
proof
To show is only for everyone . This is done by means of complete induction . The beginning of induction is trivial. Provided that the identity is valid for , this identity can be differentiated. It turns out
Applying the induction hypothesis implies
Conclusion: construction of a fundamental system
The characteristic equation for the differential equation of is
Now denote the zeros of the characteristic polynomial and the multiplicity of , so forms
a fundamental system of the equation for . So is
a fundamental system of the (homogeneous) Euler differential equation.
example
Euler's differential equation is given
According to the above sentence, the following linear differential equation with constant coefficients has to be solved
so
The characteristic polynomial belonging to this differential equation is
and has the zeros
Case 1 :, both real.
Then there is a fundamental system for the transformed linear differential equation. The inverse transformation yields that there is
a fundamental system for the original Euler's differential equation.
Case 2: .
Then there is a double zero of the characteristic polynomial. Hence is a fundamental system for the transformed linear differential equation. The inverse transformation yields that there is
a fundamental system for the original Euler's differential equation.
Case 3: both not real.
Then are complex conjugate to each other. So is a (complex) fundamental system. Be , . Then is a real fundamental system of the transformed linear differential equation. Inverse transformation provides the fundamental system for the original Euler's differential equation.
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