Cauchy-Euler Equations

Core Idea

A Cauchy-Euler (or equidimensional) equation has the form ax²y'' + bxy' + cy = 0, where each derivative is multiplied by the matching power of x. It has variable coefficients, yet it can be solved exactly. The trial solution y = xʳ turns it into the indicial equation ar(r - 1) + br + c = 0. The roots give the solutions, with three cases parallel to constant-coefficient equations: distinct real roots, a repeated root (which brings in ln x), and complex roots (which bring in cos(β ln x) and sin(β ln x)).

How It's Best Learned

Have students substitute y = xʳ and watch every term become a multiple of xʳ, which explains why powers of x are the right guess. Then show that the change of variable x = eᵗ converts the equation into one with constant coefficients, which explains the ln x terms. Work one example of each root case and check each answer by substitution.

Common Misconceptions

• The characteristic equation is ar² + br + c = 0, as for constant coefficients. (The indicial equation is ar(r - 1) + br + c = 0; the extra -ar term comes from differentiating xʳ twice.)
• A repeated root r gives solutions xʳ and x·xʳ. (The second solution is xʳ ln x, since x = eᵗ turns the constant-coefficient pattern teʳᵗ into xʳ ln x.)
• The solutions are valid for all real x. (The equation is singular at x = 0, and the formulas are written for x > 0; for x < 0 replace x by |x|.)

Explainer

Most second-order linear equations with variable coefficients cannot be solved with elementary functions. The Cauchy-Euler equation

ax²y'' + bxy' + cy = 0

is an important exception. Each derivative is multiplied by the power of x that matches its order. That pattern suggests a trial solution. If y = xʳ, then xy' = r xʳ and x²y'' = r(r - 1) xʳ. Every term is a multiple of xʳ. Substituting and dividing by xʳ (for x > 0) gives the indicial equation

ar(r - 1) + br + c = 0.

The solutions depend on the roots, in three cases that mirror constant-coefficient equations.

Distinct real roots r₁ and r₂ give y = c₁x^r₁ + c₂x^r₂. For x²y'' - 2xy' + 2y = 0, the indicial equation is r² - 3r + 2 = 0, with roots 1 and 2, so y = c₁x + c₂x².

A repeated root r gives only one power solution, xʳ. The second solution is xʳ ln x, found by reduction of order. For x²y'' + 3xy' + y = 0, the indicial equation is (r + 1)² = 0, so y = (c₁ + c₂ ln x)x⁻¹.

Complex roots α ± βi give x^{α + βi} = x^α e^{iβ ln x}. Taking real and imaginary parts yields y = x^α(c₁ cos(β ln x) + c₂ sin(β ln x)). For x²y'' + xy' + y = 0, the roots are ±i and y = c₁ cos(ln x) + c₂ sin(ln x). These solutions oscillate, with the oscillations spreading out as x grows, because ln x grows slowly.

Why do logarithms appear? Substitute x = eᵗ, so t = ln x. The chain rule gives x dy/dx = dy/dt and x² d²y/dx² = d²y/dt² - dy/dt. The equation becomes ay'' + (b - a)y' + cy = 0 in the variable t, which has constant coefficients. Its characteristic equation is the indicial equation. The familiar solutions eʳᵗ and teʳᵗ translate back to xʳ and xʳ ln x.

The coefficient x² vanishes at x = 0, so x = 0 is a singular point. The formulas above hold for x > 0, and replacing x by |x| covers x < 0.

Cauchy-Euler equations are the model case for the method of Frobenius. Near a regular singular point, a general equation behaves like a Cauchy-Euler equation, and its indicial equation is found the same way.