Solution to Differential Equation/Examples/Arbitrary Order 1 ODE: 1

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


 * $y = \ln x + C$

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

Then $\map f x$ is a solution to the first order ODE:
 * $(1): y' = \dfrac 1 x$

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 for those values of $x$ the logarithm is not defined.

It is also noted that $(1)$ is indeed defined for all $x \in \R_{>0}$.

Having established that, we continue:

and it is seen immediately that $(2)$ is the first order ODE $(1)$.