Cauchy–Euler equation

Definition

Let $n \ge 1$ and let $a_0, \dots, a_n$ be real numbers with $a_n \ne 0$. The Cauchy–Euler equation with these coefficients is the linear differential equation $$a_n x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \dots + a_1 x\, y' + a_0\, y = 0$$ for an unknown function $y$ on the interval $x > 0$.

Remarks

The substitution $x = e^{t}$, $u(t) = y(e^t)$ changes it into a linear equation with constant coefficients for $u$. For $n = 2$, the trial solution $y = x^m$ gives the indicial equation $a_2 m (m - 1) + a_1 m + a_0 = 0$. For order $2$: a double root $m$ gives the solutions $x^m$ and $x^m \ln x$. Roots $\alpha \pm i \beta$ with $\beta \ne 0$ give $x^\alpha \cos(\beta \ln x)$ and $x^\alpha \sin(\beta \ln x)$. Example: $x^2 y'' - 2 y = 0$ gives $m(m-1) - 2 = 0$, so $m = 2$ or $m = -1$, and the solutions are $c_1 x^2 + c_2 x^{-1}$.

Explanations

L1 human

In this equation, each derivative $y^{(k)}$ is multiplied by $x^k$. So powers $x^m$ behave like exponentials do for constant coefficients: the equation becomes a polynomial equation in $m$.