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

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## 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 $(1)$.

$\blacksquare$

## Sources

- 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: Example $3.51$