Solution to Differential Equation/Examples/Absolute Value Function

Examples of Solutions to Differential Equations

Consider the real function defined as:

$\map f x = \size x$

where $\size x$ is the absolute value function.

Then $\map f x$ cannot be the solution to a differential equation.

However, by suitably restricting $\map f x$ to a domain which does not include $x = 0$, there may well exist differential equations for which the resulting real function is a solution.

Proof

By definition, for $\map f x$ to be a solution to a differential equation $Q$, it must satisfy $Q$ at all elements of its domain.

But by Derivative of Absolute Value Function, the derivative of $\size x$ with respect to $x$ does not exist at $x = 0$.

So at $x = 0$, $\map {f'} x$ is not defined.

However,let $\map f x$ be restricted to $x \ne 0$:

$\forall x \in \R_{\ne 0}: \map g x := \size x$

Then it is seen that $\map g x$ is a solution to the differential equation:

$y' = \begin {cases} 1 & : x > 0 \\ -1 & : x < 0 \end {cases}$

$\blacksquare$

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: Comment $3.53$