[tex]x^2y''+xy'-y &=2x\\
x^2y''+2xy'-xy'-y &=2x\\
(x^2y')'-(xy)' &=2x\\
(x^2y'-xy)' &=2x\\
x^2y'-xy &=x^2+C_1\\
xy'-y &=x+\frac{C_1}{x}\\
x^2\left(\frac{y}{x}\right)' &=x+\frac{C_1}{x}\\
\left(\frac{y}{x}\right)' &=\frac{1}{x}+\frac{C_1}{x^3}\\
\frac{y}{x} &=\ln{x}-\frac{C_1}{2x^2}+C_2\\
y &=x\ln{x}-\frac{C_1}{2x}+C_2x[/tex]
