Cauchy-Euler equation
/ KOH-shee OY-ler /
Most variable-coefficient ODEs are genuinely hard, but one special pattern stays almost as easy as the constant-coefficient case. Whenever each derivative is multiplied by a matching power of x — x^2 y'' with x^2, x y' with x, plain y — the equation is a Cauchy-Euler equation, and it yields to a single clean trick.
The standard form is a x^2 y'' + b x y' + c y = 0. Because the powers of x exactly balance the orders of differentiation, the trial solution y = x^m turns each term into a constant times x^m. Substituting gives the indicial equation a m(m-1) + b m + c = 0, a quadratic in m whose roots give the solutions, in close parallel to the characteristic equation: distinct real roots m_1, m_2 give c_1 x^(m_1) + c_2 x^(m_2); a repeated root m gives (c_1 + c_2 ln x) x^m; a complex pair alpha plus-or-minus i beta gives x^alpha (c_1 cos(beta ln x) + c_2 sin(beta ln x)). Equivalently, the substitution x = e^t converts it exactly into a constant-coefficient equation in t.
Cauchy-Euler equations are not a curiosity: they are precisely what you meet after separating Laplace's equation in polar, cylindrical, or spherical coordinates — the radial part is Cauchy-Euler. So they govern steady temperature, electrostatic potential, and fluid flow in radially symmetric regions, and they are the gateway to recognizing x = 0 as a regular singular point handled more generally by the Frobenius method.
x^2 y'' - 2x y' - 4y = 0: try y = x^m, giving m(m-1) - 2m - 4 = m^2 - 3m - 4 = 0, roots m = 4 and m = -1. So y = c_1 x^4 + c_2 x^(-1).
Powers x^m play for Cauchy-Euler the role that exponentials e^(rx) play for constant coefficients.
The solutions x^m have a singularity at x = 0 (for negative or fractional m), so a Cauchy-Euler solution is normally valid only on x > 0 or x < 0, not across the origin; choose the interval to match the problem.