Cauchy Euler Equations: Concepts, Formulas and Examples

A Cauchy–Euler equation is a linear ODE where each derivative is multiplied by a matching power of x, like x²y″ + axy′ + by = 0. The powers of x and the orders of the derivatives balance so that trying y = x^m reduces the whole equation to algebra, just as y = e^{rx} does for constant coefficients.

These equations matter because they appear wherever radial symmetry or scaling matters: vibrations of circular membranes, fluid flow in wedges, and the radial part of Laplace's equation. They also connect to constant-coefficient equations through the substitution t = ln x.

Key Concepts

• Form. x²y″ + axy′ + by = 0 (second order); generally x^n y^{(n)} with matching powers.
• Trial solution. y = x^m gives y′ = mx^{m−1} and y″ = m(m − 1)x^{m−2}, so each term becomes a multiple of x^m.
• Characteristic equation. m(m − 1) + am + b = 0, a quadratic for m.
• Repeated root. If m1 = m2 = m, the second solution is x^m ln x, mirroring the xe^{rx} rule.
• Complex roots. m = α ± iβ gives x^α cos(β ln x) and x^α sin(β ln x).

Formulas

• Characteristic: for x²y″ + axy′ + by = 0, solve m² + (a − 1)m + b = 0.
• Distinct real roots: y = C1x^{m1} + C2x^{m2}.
• Repeated root: y = (C1 + C2 ln x)x^m.
• Substitution link: with t = ln x (so x = e^t), the equation becomes a constant-coefficient ODE in t, valid for x > 0.
• Assumptions and limits: ln x requires x > 0; for x < 0 use ln|x|. The method needs the exact power-derivative matching of the Cauchy–Euler form.

Worked Example

Solve x²y″ − xy′ + y = 0 for x > 0.

1. Try y = x^m: y′ = mx^{m−1}, y″ = m(m − 1)x^{m−2}.
2. Substitute: x²·m(m − 1)x^{m−2} − x·mx^{m−1} + x^m = [m(m − 1) − m + 1]x^m = 0.
3. Characteristic equation: m(m − 1) − m + 1 = m² − 2m + 1 = (m − 1)² = 0, so m = 1 is a double root.
4. Repeated root rule: y = (C1 + C2 ln x)x^1 = C1x + C2x ln x.
5. Verify the second piece y2 = x ln x: y2′ = ln x + 1 (product rule) and y2″ = 1/x.
6. Substitute: x²(1/x) − x(ln x + 1) + x ln x = x − x ln x − x + x ln x = 0. Correct.

Common Misconceptions

• Using e^{rx} as the trial. Exponentials suit constant coefficients; Cauchy–Euler equations need the power trial y = x^m.
• Forgetting the ln x on repeated roots. Just as repeated r needs xe^{rx}, repeated m needs x^m ln x for the second solution.
• Ignoring the domain. ln x is only defined for x > 0; the formula with ln|x| extends the solution to x < 0.

Real-World Applications

• Polar and spherical problems: the radial equation in Laplace's equation separates into Cauchy–Euler form.
• Elasticity: stress distributions in wedges and around holes involve Euler-type equations.
• Emden–Fowler equations: stellar structure models reduce to related equidimensional forms.

Cauchy–Euler equations balance powers of x against derivative orders, so the trial y = x^m turns them into the quadratic m² + (a − 1)m + b = 0. The double root m = 1 in x²y″ − xy′ + y = 0 gives y = (C1 + C2 ln x)x, verified term by term.