Solve x²y''−xy'+2y=x log x

x²y'' − xy' + 2y = x log x is a Cauchy–Euler equation.

Transformation

Put z = log x, D = d/dz. The right side becomes x log x = z eᶻ.

[D(D − 1) − D + 2]y = z eᶻ (D² − 2D + 2)y = z eᶻ

Complementary function

m² − 2m + 2 = 0, so m = 1 ± i.

CF = eᶻ(c₁ cos z + c₂ sin z) = x[c₁ cos(log x) + c₂ sin(log x)]

Particular integral

Replace D by D + 1:

PI = eᶻ · 1/[(D + 1)² − 2(D + 1) + 2] · z = eᶻ · 1/(D² + 1) · z = eᶻ(1 − D² + …) z = z eᶻ

General solution

y = x[c₁ cos(log x) + c₂ sin(log x)] + x log x