Second Order ODE/x^2 y'' + x y' = 1

Theorem
The second order ODE:
 * $x^2 y'' + x y' = 1$

has the general solution:
 * $y = \dfrac {\paren {\ln x}^2} 2 + C_1 \ln x + C_2$

Proof
The proof proceeds by using Solution of Second Order Differential Equation with Missing Dependent Variable.

Substitute $p$ for $y'$: