# 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:

 $\ds y$ $=$ $\ds \ln x + C$ $\text {(2)}: \quad$ $\ds \leadsto \ \$ $\ds y'$ $=$ $\ds \dfrac 1 x$ Derivative of Natural Logarithm, Derivative of Constant

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

$\blacksquare$