= Cauchy-Euler equation
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{wiki=Cauchy–Euler_equation}
= Cauchy-Euler differential equation
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{synonym}
= Euler differential equation
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A <Cauchy-Euler equation> is a <linear ordinary differential equation> whose coefficient powers match their derivative orders, for example
$$
x^2y''+a x y'+b y=f(x),\qquad x>0.
$$
With $t=\log x$ and $Y(t)=y(e^t)$, the <chain rule> gives $xy'=Y_t$ and $x^2y''=Y_{tt}-Y_t$. Thus the equation becomes the <constant-coefficient differential equation> $Y_{tt}+(a-1)Y_t+bY=f(e^t)$. In the homogeneous case, the <power-law ansatz> $y=x^r$ gives the <indicial equation> $r(r-1)+ar+b=0$.
For $x^2y''-xy'+y=0$, the repeated root $r=1$ yields $y=x(C_1+C_2\log x)$. The logarithm is the transformed repeated-root factor $t e^t$. Negative real $x$ can be handled separately using $t=\log|x|$, while complex powers require a chosen logarithm branch. The singular point at zero is not crossed automatically by the logarithmic coordinate.
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