---
tag: 0T6D4PE51C
kind: example
name: Clairaut's equation
msc: [34A09]
instance_of: [08WWTN738R]
uses: [0CZA0NCFWP, 0SKYFQAFYD]
sources:
  - url: https://en.wikipedia.org/wiki/Clairaut%27s_equation
    locator: Solution
same_as:
  wikidata: Q1094772
  wikipedia: Clairaut's equation
---

## Example

Let $f$ be a twice continuously differentiable function on an open interval $J$.
Clairaut's equation is the implicit first-order equation
$$y = x\, y' + f(y'),$$
for an unknown $y$ with $y' \in J$.
For each constant $C \in J$, the line $y = C x + f(C)$ is a solution. These lines form the [[general solution|0SKYFQAFYD]].
If $f'' \ne 0$ on $J$, the curve with the parametric form $x = -f'(p)$, $y = f(p) - p f'(p)$ for $p \in J$ is also a solution. It is the envelope of the lines, and it is a [[singular solution|0CZA0NCFWP]].

## Remarks

Differentiation of the equation gives $y''\left(x + f'(y')\right) = 0$. The case $y'' = 0$ gives the lines, and the case $x + f'(y') = 0$ gives the envelope.
Example: for $f(p) = p^2$, the lines are $y = C x + C^2$, and the singular solution is the parabola $y = -x^2/4$.
Each line of the general solution is tangent to the singular solution at one point.

## Explanations

### L1 human

Clairaut's equation has a family of straight lines as solutions, one for each slope $C$.
These lines wrap around a curve, their envelope, and this curve is also a solution.
No choice of $C$ gives the envelope, so it is a new, singular solution.