Solution to Differential Equation/Examples/Arbitrary Order 2 ODE

Examples of Solutions to Differential Equations
Consider the real function defined as:


 * $y = \map f x = \ln x + x$

defined on the domain $x \in \R_{>0}$.

Then $\map f x$ is a solution to the second order ODE:
 * $(1): \quad x^2 y'' + 2 x y' + y = \ln x + 3 x + 1$

defined on the domain $x \in \R_{>0}$.

Proof
It is noted that $\map f x$ is not defined in $\R$ when $x \le 0$ because that is outside the domain of the natural logarithm $\ln$.

The same constraint applies to $(1)$.

Having established that, we continue:

Then: