Solve x²y''−4xy'+6y=x

x²y'' − 4xy' + 6y = x is a Cauchy–Euler equation.

Transformation

Put z = log x, θ = d/dz: x²y'' = θ(θ − 1)y and xy' = θy. The right side is x = eᶻ.

[θ(θ − 1) − 4θ + 6]y = eᶻ (θ² − 5θ + 6)y = eᶻ, i.e. (θ − 2)(θ − 3)y = eᶻ

Complementary function

m = 2, 3: CF = c₁e^(2z) + c₂e^(3z) = c₁x² + c₂x³

Particular integral

Put θ = 1: PI = eᶻ/(1 − 5 + 6) = eᶻ/2 = x/2

General solution

y = c₁x² + c₂x³ + x/2