Cauchy–Euler equation

In mathematics, an Euler–Cauchy equation, also known as a Cauchy–Euler equation, equidimensional equation, or Euler's equation, is a linear ordinary differential equation for which the homogeneous part is invariant under changes

to the scale of its independent variable. Euler was

the first we know of to study equations of this form in the early 1700's, with a notable appearance in

Institutiones calculi integralis, volume 2 in 1768.

The equation

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

a

n

x

n

y

(

n

)

(

x

)

+

a

n

−

1

x

n

−

1

y

(

n

−

1

)

(

x

)

+

⋯

+

a

0

y

(

x

)

=

0.

{\displaystyle a_{n}x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+\dots +a_{0}y(x)=0.}

The substitution

x

=

e

u

{\displaystyle x=e^{u}}

(that is,

u

=

ln

⁡

(

x

)

{\displaystyle u=\ln(x)}

; for

x

<

0

{\displaystyle x<0}

, in which one might replace all instances of

x

{\displaystyle x}

by

|

x

|

{\displaystyle |x|}

, extending the solution's domain to

R

∖

{

0

}

{\displaystyle \mathbb {R} \setminus \{0\}}

) can be used to reduce this equation to a linear differential equation with constant coefficients. Alternatively, the trial solution

y

=

x

m

{\displaystyle y=x^{m}}

can be used to solve the equation directly, yielding the basic solutions.

Second order – solving through trial solution

The most common Cauchy–Euler equation is the second-order equation, which appears in a number of physics and engineering applications, such as when solving Laplace's equation in polar coordinates. The second order Cauchy–Euler equation is

x

2

d

2

y

d

x

2

+

a

x

d

y

d

x

+

b

y

=

0.

{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+ax{\frac {dy}{dx}}+by=0.}

We assume a trial solution

y

=

x

m

.

{\displaystyle y=x^{m}.}

Differentiating gives

d

y

d

x

=

m

x

m

−

1

{\displaystyle {\frac {dy}{dx}}=mx^{m-1}}

and

d

2

y

d

x

2

=

m

(

m

−

1

)

x

m

−

2

.