Cauchy-Euler equation

A Cauchy–Euler equation has powers of the independent variable matched to derivative order and is solved using power laws or a logarithmic change of variable.

Normalize the second-order equation as $y^{\prime \prime }+P(z)y^{\prime }+Q(z)y=0$. If every point other than zero and infinity is ordinary and both exceptions are at worst regular singular points, then $zP$ and $z^{2}Q$ extend holomorphically across zero and stay bounded at infinity. Liouville's theorem makes them constants, yielding the displayed Euler-Cauchy equation. If both exceptions must be genuinely singular, exclude $(A,B)=(0,0)$ and $(2,0)$, which make zero and infinity respectively ordinary.

Forthe power ansatz $w=z^{\lambda }$ gives the indicial equation $\lambda (\lambda -1)+a\lambda +b=0$. Distinct roots give $C_1{z}^{{\lambda }_1}+C_2{z}^{{\lambda }_2}$; a repeated root $\lambda$ gives $z^{\lambda }(C_1+C_2\log z)$ on a chosen logarithm branch.

A power-law ansatz assumes that an unknown function is a monomial $y=x^s$. Scale-invariant differential equations then reduce to an algebraic equation for the exponent $s$.

An indicial equation is the algebraic equation for the leading exponent of a power-law or Frobenius solution near a regular singular point.

An indicial root is a root of the indicial equation obtained from a power-law ansatz. For a Cauchy-Euler differential equation, such roots give the exponents of its power solutions. Their sign determines which modes remain bounded near a singular endpoint.

An indicial exponent is a root of the indicial equation at a regular singular point. It gives the leading local power in a Frobenius method; resonant roots can require logarithmic terms.

A log-periodic oscillation is periodic in the logarithm of its argument, for example $\cos (\mu \log x)$. Its successive zeros or extrema therefore occur at radii in a geometric progression.