Clairaut's equation

To solve Clairaut's equation, one differentiates with respect to $x$, yielding

:$\backslash frac\{dy\}\{dx\}=\backslash frac\{dy\}\{dx\}+x\backslash frac\{d^2\; y\}\{dx^2\}+f\text{'}\backslash left(\backslash frac\{dy\}\{dx\}\backslash right)\backslash frac\{d^2\; y\}\{dx^2\},$

so

:$\backslash left[x+f\text{'}\backslash left(\backslash frac\{dy\}\{dx\}\backslash right)\backslash right]\backslash frac\{d^2\; y\}\{dx^2\}\; =\; 0.$

Hence, either

:$\backslash frac\{d^2\; y\}\{dx^2\}\; =\; 0$

or

:$x+f\text{'}\backslash left(\backslash frac\{dy\}\{dx\}\backslash right)\; =\; 0.$

In the former case, $C\; =\; dy/dx$ for some constant $C$. Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by

:$y(x)=Cx+f(C),\backslash ,$

the so-called general solution of Clairaut's equation.

The latter case,

:$x+f\text{'}\backslash left(\backslash frac\{dy\}\{dx\}\backslash right)\; =\; 0,$

defines only one solution $y(x)$, the so-called singular solution, whose graph is the envelope of the graphs of the general solutions. The singular solution is usually represented using parametric notation, as $(x(p),\; y(p))$, where $p\; =\; dy/dx$.

The parametric description of the singular solution has the form

:$x(t)=\; -f\text{'}(t),\backslash ,$

:$y(t)=\; f(t)\; -\; tf\text{'}(t),\backslash ,$

where $t$ is a parameter.

The following curves represent the solutions to two Clairaut's equations:

Image:Solutions to Clairaut's equation where f(t)=t^2.png|{{center|1=$f(p)=p^2$}}

Image:Solutions to Clairaut's equation where f(t)=t^3.png|{{center|1=$f(p)=p^3$}}

In each case, the general solutions are depicted in black while the singular solution is in violet.

By extension, a first-order partial differential equation of the form

:$\backslash displaystyle\; u=xu\_x+yu\_y+f(u\_x,u\_y)$

is also known as Clairaut's equation.{{harvnb|Kamke|1944}}.

{{Portal|Mathematics}}

• D'Alembert's equation
• Chrystal's equation
• Legendre transformation

{{Reflist}}

• {{Citation

| last = Clairaut

| first = Alexis Claude

| title = Solution de plusieurs problèmes où il s'agit de trouver des Courbes dont la propriété consiste dans une certaine relation entre leurs branches, exprimée par une Équation donnée.

| url = http://gallica.bnf.fr/ark:/12148/bpt6k3531x/f344.table

| journal = Histoire de l'Académie Royale des Sciences

| year = 1734

| pages = 196–215

}}.

• {{Citation

| last = Kamke

| first = E.

| language = de

| title = Differentialgleichungen: Lösungen und Lösungsmethoden

| volume = 2. Partielle Differentialgleichungen 1er Ordnung für eine gesuchte Funktion

| publisher = Akad. Verlagsgesell

| year = 1944

}}.

• {{springer

| title = Clairaut equation

| id = C/c022350

| last = Rozov

| first = N. Kh.

}}.

Category:Eponymous equations of mathematics

Category:Ordinary differential equations