Cauchy–Euler equation

CauchyEuler equation

1

Cauchy €Euler equation In mathematics, a Cauchy €Euler equation (also known as the Euler €Cauchy equation, equation, or simply Euler's equation) equation) is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its simple structure the equation can be replaced with an equivalent equation with constant coefficients which can then be solved explicitly.

The equation Let  y

(n)

( x  x) be the nth derivative of the unknown function y( x  x). Then a Cauchy €Euler equation of order n has the

form

The substitution

reduces this equation to a linear differential equation with constant coefficients.

Alternatively a trial solution

[1]

may be used to solve for the basis solutions.

Second order - solving through trial solution The most common Cauchy-Euler equation is the second-order equation, appearing in a number of physics and engineering applications, such as when solving Laplace's equation in polar coordinates. It is given by the equation

[1]

:

Typical solution curves for a second-order Euler € Cauchy Cauchy equation for the case of two real roots

Typical solution curves for a second-order Euler € Cauchy Cauchy equation for the case of a double root

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 and m 1

2

• 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

m

To get to this solution, the method of reduction of order must be applied after having found one solution  y = x . In case #3, the solution is given by:

For

and

in the real plane t

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

CauchyEuler equation

3

Second order - solution through change of variables

We operate the variable substitution defined by

Differentiating:

Substituting :

This equation in

Now, if

and

, we have

can be easily solved using its characteristic polynomial

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 Given

‚

we 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  •  (n) m

as

Applying the difference operator to

, we find that

If we do this k times, we will find that

where the superscript

(k )

denotes applying the difference operator k times. Comparing this to the fact that the k -th

m

derivative of x 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  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 m  will yield expressions involving a discrete version of ln, 1

(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.

Article Sources and Contributors

Article Sources and Contributors Cauchy €Euler equation Source: http://en.wikipedia.org/w/index.php?oldid=455028494 Contributors: Amakthea computer, Brrk.3001, Burn, Cadillac, Danski14, Dysprosia, Flyrev, FrankTobia, Gidons, Gscshoyru, Hauberg, JB Gnome, JJ Harrison, Jiy, Kiensvay, Linas, Lowe4091, MathMartin, Matt me, Mets501, Michael Hardy, NabiNabiya, Oleg Alexandrov, Qcosmos, Salgueiro, Staecker, Svick, Titoxd, Vanished User 0001, Vivacissamamente, Wikieditor06, 29 anonymous edits

Image Sources, Licenses and Contributors File:Euler-Cauchy equation solution curves real roots.png Source: http://en.wikipedia.org/w/index.php?title=File:Euler-Cauchy_equation_solution_curves_real_roots.png License: Public Domain Contributors: Noodle snacks File:Euler-Cauchy equation solution curves double root.png Source: http://en.wikipedia.org/w/index.php?title=File:Euler-Cauchy_equation_solution_curves_double_root.png License: Public Domain Contributors: Noodle snacks File:Euler-Cauchy equation solution curves complex roots.png Source: http://en.wikipedia.org/w/index.php?title=File:Euler-Cauchy_equation_solution_curves_complex_roots.png License: Public Domain Contributors: Noodle snacks

License Creative Commons Attribution-Share Alike 3.0 Unported  //creativecommons.org/licenses/by-sa/3.0/ 

5