Let
Let y y((nn))(( x x) be the) be the nnth derivative of the unknown functionth derivative of the unknown function y y(( x x). Then a). Then a CauchyCauchy € €Euler equationEuler equation of orderof order nn has thehas the
form form
The
The substitution substitution reduces reduces this this equation equation to to a a linear linear differential differential equation equation with with constant constant coefficients.coefficients. Alternatively
Alternatively a a trial trial solution solution may may be be used used to to solve solve for for the the basis basis solutions.solutions.[1][1]
Second order - solving through trial solution
Second order - solving through trial solution
Typical solution curves for a second-order Typical solution curves for a second-order Euler
Euler € € Cauchy equation for the case of two realCauchy equation for the case of two real
roots roots
Typical solution curves for a second-order Typical solution curves for a second-order Euler
Euler € € Cauchy equation for the case of a doubleCauchy equation for the case of a double
root root
The most common Cauchy-Euler equation is the second-order The most common Cauchy-Euler equation is the second-order equation, appearing in a number of physics and engineering equation, appearing in a number of physics and engineering applications, such as when solving Laplace's equation in polar applications, such as when solving Laplace's equation in polar coordinates. It is given by the equation
CauchyEuler equation 2
Typical solution curves for a second-order Euler € Cauchy equation for the case of complex
roots
We assume a trial solution given by[1]
Differentiating, we have:
and
Substituting into the original equation, we have:
Or rearranging gives:
We then can solve for m. There are three particular cases of interest: • Case #1: Two distinct roots,m
1andm2
• Case #2: One real repeated root,m
• Case #3: Complex roots, ‚ ƒ „i
In case #1, the solution is given by:
In case #2, the solution is given by
To get to this solution, the method of reduction of order must be applied after having found one solution y = xm. In case #3, the solution is given by:
For and in the real plane
Substituting : , we have
This equation in can be easily solved using its characteristic polynomial
Now, if and are the roots of this polynomial, we analyze the two main cases: distinct roots and double roots: If the roots are distinct, the general solution is given by
, where the exponentials may be complex. If the roots are equal, the general solution is given by
In both cases, the solution may be found by setting , hence . Hence, in the first case,
, and in the second case,
Example
Givenwe substitute the simple solution x‚:
For x‚ to be a solution, either x = 0, which gives the trivial solution, or the coefficient of x‚ is zero. Solving the quadratic equation, we get € = 1, 3. The general solution is therefore
CauchyEuler equation 4
Difference equation analogue
There is a difference equation analogue to the Cauchy € Euler equation. For a fixed m > 0, define the sequence • m(n)
as
Applying the difference operator to , we find that
If we do thisktimes, we will find that
where the superscript (k )denotes applying the difference operator k times. Comparing this to the fact that the k -th derivative of xmequals
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 N -th order linear difference equation is also the linear combination of Nlinearly independent solutions. Applying reduction of order in case of a multiple rootm
1 will yield expressions involving a discrete version of ln,
(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 m.
References
[1] Kreyszig, Erwin (May 10, 2006). Advanced Engineering Mathematics. Wiley. ISBN 978-0470084847.
Bibliography
• Weisstein, Eric W., " Cauchy € Euler equation (http:/ / mathworld.wolfram.com/ EulerDifferentialEquation. html)" from MathWorld.
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