Cauchy–Euler Equations

Differential Equations · Axiom Academy

When the coefficients grow with x in step with the derivative order, guess a power instead of an exponential. A second-order Cauchy–Euler equation looks like this: a , b , c constant — the powers of x are what vary The signature is unmistakable once you know it: the power of x in each coefficient matches the order of the derivative it multiplies . Two derivatives get x^2 , one derivative gets x^1 , no derivatives get x^0 = 1 . That pairing is not cosmetic — it is precisely what makes the equation solvable. Watch why. Differentiating a power lowers its exponent; each x^k coefficient raises it back by exactly the amount that was lost. So every term emerges carrying the same power x^ r , which can then be divided out. That common factor is the entire mechanism. Substituting y = x^ r and cancelling the common x^ r leaves a plain quadratic in r — the indicial equation . It plays exactly the role the characteristic equation plays for constant coefficients. Note the r(r-1) , not r^2 — that comes from differentiating x^r twice. Expanded, it reads ar^2 + (b-a)r + c = 0 . In the animation, r is swept continuously: the curve y = x^ r bends as it goes, and the residual on the right tracks how badly that r fails. The curve locks in exactly when the residual hits zero. The equation being searched is the one the course tests you on: Two distinct real roots, so two independent power solutions, and the general solution is their combination:

This is the written version of the interactive lesson above. See the full Differential Equations course.