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

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

has the solution:
 * $y = - \dfrac {x^2} 2 - C_1 x - C_1^2 \ln \left({x - C_1}\right) + C_2$

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

Substitute $p$ for $y'$: