# 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:

 $\ds y$ $=$ $\ds \ln x + x$ $\ds \leadsto \ \$ $\ds y'$ $=$ $\ds \dfrac 1 x + 1$ Derivative of Natural Logarithm, Derivative of Identity Function $\ds \leadsto \ \$ $\ds y''$ $=$ $\ds -\dfrac 1 {x^2} + 0$ Power Rule for Derivatives, Derivative of Constant

Then:

 $\ds$  $\ds x^2 \paren {-\dfrac 1 {x^2} } + 2 x \paren {\dfrac 1 x + 1} + \ln x + x$ substituting for $y$, $y'$ and $y''$ from above into the left hand side of $(1)$ $\ds$ $=$ $\ds -1 + 2 + 2 x + \ln x + x$ $\ds$ $=$ $\ds \ln x + 3 x + 1$ which equals the right hand side of $(1)$

$\blacksquare$