Solve the Second Order Nonhomogeneous ODE I'm stuck on this problem since I can't find any material in my textbook or internet to solve this. Any help?
$$2x^2 \cdot y'' - x \cdot y' + y = x$$
$$y'' - \frac{y'}{2x} + \frac{y}{2x^2} = \frac{1}{2x}$$

Here is another way:
$$y'' - \frac{y'}{2x} + \frac{y}{2x^2} = \frac{1}{2x}$$
$$y'' -\frac 1 2 ( \frac{y'x}{x^2} - \frac{y}{x^2})= \frac{1}{2x}$$
$$y'' -\frac 1 2 ( \frac{y'x-y}{x^2})= \frac{1}{2x}$$
$$y'' -\frac 1 2 ( \frac{y}{x})'= \frac{1}{2x}$$
Integrate:
$$y' -\frac{y}{2x}=\int \frac{dx}{2x}$$
$$y' -\frac{y}{2x}=\frac{\ln(x)}{2}+K$$
Which is easy to solve ....
