Cauchy-Euler equation

In mathematics, a Cauchy-Euler equation (also Euler-Cauchy equation) is a linear homogeneous ordinary differential equation with variable coefficients. They are sometimes known as equi-dimensional equations. 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 $$y^{(n)}(x) be the "n"th derivative of the unknown function $$y(x). Then a Cauchy-Euler equation of order "n" has the form

:$$x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + cdots + a_0 y(x) = 0.

The substitution $$scriptstyle x = e^u reduces this equation to a linear differential equation with constant coefficients.

Second order

The Euler-Cauchy equation crops up in a number of engineering applications. It is given by the equation:

:$$x^2frac{d^2y}{dx^2} + axfrac{dy}{dx} + by = 0 ,

We assume a trial solution given by

:$$y = x^m. ,

Differentiating, we have:

:$$frac{dy}{dx} = mx^{m-1} ,

and

:$$frac{d^2y}{dx^2} = m(m-1)x^{m-2}. ,

Substituting into the original equation, we have:

:$$x^2( m(m-1)x^{m-2} ) + ax( mx^{m-1} ) + b( x^m ) = 0 ,

Or rearranging gives:

:$$m^2 + (a-1)m + b = 0. ,

We then can solve for "m". There are three particular cases of interest:

* Case #1: Two distinct roots, "m"₁ and "m"₂
* Case #2: One real repeated root, "m"
* Case #3: Complex roots, α ± "i"β

In case #1, the solution is given by::$$y = c_{1}x^{m_{1 + c_{2}x^{m_{2 ,

In case #2, the solution is given by::$$y = c_{1}x^{m}ln(x) + c_{2}x^{m} ,To get to this solution, the method of reduction of order must be applied after having found one solution "y" = "x"^"m".

In case #3, the solution is given by:

:$$y = c_{1}x^{alpha}cos(eta ln(x)) + c_{2}x^{alpha}sin(eta ln(x)) ,

This equation also can be solved with $$x = e^t transformation.

This particular case is of no great practical importance and hence this has been left as a challenge for the reader.

Example

Given : $$x^2u"-3xu'+3u=0,we substitute the simple solution "x"^α:: $$x^2(alpha(alpha-1)x^{alpha-2})-3x(alpha(x^{alpha-1}))+3(x^alpha)=alpha(alpha-1)x^alpha-3alpha x^alpha+3x^alpha,: $$(alpha(alpha-1)-3alpha+3)x^alpha.,For this to indeed be a solution, either "x"=0 giving the trivial solution, or the coefficient of "x"^α is zero, so solving that quadratic, we get α=1,3. So, the general solution is: $$u=Ax+Bx^3.,

Difference equation analogue

There is a difference equation analogue to the Cauchy–Euler equation. For a fixed $$m > 0, define the sequence $$f_m(n) as

: $$f_m(n) := n(n+1)cdots (n+m-1).

Applying the difference operator to $$f_m, we find that

: $$Df_m(n) = f_{m+1}(n) - f_m(n) = m(n+1)(n+2) cdots (n+m-1) = frac{m}{n} f_m(n).

If we do this "k" times, we will find that

: $$f_m^{(k)}(n) = frac{m(m-1)cdots(m-k+1)}{n(n+1)cdots(n+k-1)} f_m(n) = m(m-1)cdots(m-k+1) frac{f_m(n)}{f_k(n)},

where the superscript ^("k") denotes applying the difference operator "k" times. Comparing this to the fact that the "k"-th derivative of $$x^m equals $$m(m-1)cdots(m-k+1)frac{x^m}{x^k} suggests that we can solve the "N"-th order difference equation

: $$f_N(n) y^{(N)}(n) + a_{N-1} f_{N-1}(n) y^{(N-1)}(n) + cdots + a_0 y(n) = 0,

in a similar manner to the differential equation case. Indeed, substituting the trial solution

: $$y(n) = f_m(n)

brings us to the same situation as the differential equation case,

: $$m(m-1)cdots(m-N+1) + a_{N-1} m(m-1) cdots (m-N+2) + cdots + a_1 m + a_0 = 0.

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 $$m_1 will yield expressions involving a discrete version of $$ln,

: $$varphi(n) = sum_{k=1}^n frac{1}{k - m_1}.

(Compare with: $$ln (x - m_1) = int_{1+m_1}^x frac{1}{t - m_1} , dt.)

In cases where fractions become involved, one may use

: $$f_m(n) := frac{Gamma(n+m)}{Gamma(n)}

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

ee also

* Hypergeometric differential equation