Definition. Equations of the form $x^n{y}^{\left(n\right)}+a_{n-1}{x}^{n-1}{y}^{(n-1)}\cdots +a_{1}x{y}^{\text{'}}+a_{0}y=r\left(x\right)$ are called Euler-Cauchy equations.

Solution of homogeneous equation

• Guess solution is of form $y\left(x\right)=x^m$, $y^{\text{'}}=m{x}^{m-1},\dots ,y^{\left(n\right)}=m(m-1)\dots (m-n+1)x^{m-n}$ and obtain

  ${\displaystyle m(m-1)\dots (m-n+1)+a_{n-1}m(m-1)\dots (m-n+2)+\cdots +a_{1}m+a_0=0,}$

  a polynomial equation in $m$.

• For $n=2$, with notation $a_1=a$, $a_0=b$

  ${\displaystyle m(m-1)+am+b=m^2+(a-1)m+b=0}$

Note. A constant-coefficient equation, $y^{\left(n\right)}+a_{n-1}{y}^{(n-1)}+\cdots +a_0=0$, is a linear combination of operators $\frac{d^k}{d{x}^k}$ for which $e^{rx}$ is an eigenfunction with eigenvalue $r^k$.

The homogeneous Euler-Cauchy equation is a linear combination of operators $x^k\frac{d^k}{d{x}^k}$ for which $x^m$ is an eigenfunction with eigenvalue $m(m-1)\dots (m-k+1)$