Definition
An Euler Equation (Cauchy-Euler) is a linear ODE with variable coefficients where the power of matches the order of the derivative:
- How to read: “The equation a x squared y double prime plus b x y prime plus c y equals zero.”
- Meaning: Equidimensional ODE—power of matches derivative order; admits power-law solutions .
Why It Matters
Euler equations model systems where growth rates are relative to current scale, providing the natural basis for power-law solutions rather than exponentials. They are critical for solving Laplace’s equation in polar coordinates and analyzing the stress in rotating mechanical disks, capturing the scaling invariance of the physical world.
Core Concepts
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The Euler (Cauchy-Euler) Equation
- How to read: “The term a x squared y double prime plus b x y prime plus c y equals zero.”
- Meaning: The powers of match derivative order—equidimensional/homogeneous under scaling of ; admits instead of exponentials.
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Indicial (Characteristic) Equation
- Substitute y = x^r → ar(r-1) + br + c = 0, which simplifies to ar² + (b - a)r + c = 0.
- How to read: “The indicial equation a r squared plus the quantity b minus a times r, plus c, equals zero.”
- Meaning: Characteristic polynomial from substituting ; its roots determine the solution form.
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Solution Forms (by root type)
- Distinct real roots r1, r2: y = c1 x^{r1} + c2 x^{r2}
- Repeated real root r: y = (c1 + c2 ln x) x^r
- Complex roots α ± iβ: y = x^α [c1 cos(β ln x) + c2 sin(β ln x)]
- How to read: “The solution y equals c one times x to the r one power plus c two times x to the r two power; or for repeated roots, the quantity c one plus c two natural log of x, multiplied by x to the r; or for complex roots, x to the alpha multiplied by the quantity c one cosine of the quantity beta natural log of x, plus c two sine of the quantity beta natural log of x.”
- Meaning: Three solution templates keyed to distinct, repeated, or complex indicial roots.
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Log substitution trick: Let x = e^z (z = ln x). Then the Euler equation becomes a constant-coefficient linear ODE in z, which you already know how to solve. Back-substitute to get the x^r solutions. This is why the method works.