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Homed'Alembert's equation

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# d’Alembert’s equation

The first order differential equation

$y=\varphi(\frac{dy}{dx})\cdot x+\psi(\frac{dy}{dx})$ |

is called d’Alembert’s differential equation; here $\varphi$ and $\psi$ mean some known differentiable real functions.

If we denote $\frac{dy}{dx}:=p$, the equation is

$y=\varphi(p)\cdot x+\psi(p).$ |

We take $p$ as a new variable and derive the equation with respect to $p$, getting

$p-\varphi(p)=[x\varphi^{{\prime}}(p)+\psi^{{\prime}}(p)]\frac{dp}{dx}.$ |

If the equation $p-\varphi(p)=0$ has the roots $p=p_{1}$, $p_{2}$, …, $p_{k}$, then we have $\frac{dp_{{\nu}}}{dx}=0$ for all $\nu$’s, and therefore there are the special solutions

$y=p_{{\nu}}x+\psi(p_{{\nu}})\quad(\nu=1,2,...,k)$ |

for the original equation. If $\varphi(p)\not\equiv p$, then the derived equation may be written as

$\frac{dx}{dp}=\frac{\varphi^{{\prime}}(p)}{p-\varphi(p)}x+\frac{\psi^{{\prime}% }(p)}{p-\varphi(p)},$ |

which linear differential equation has the solution $x=x(p,C)$ with the integration constant $C$. Thus we get the general solution of d’Alembert’s equation as a parametric representation

$\begin{cases}x=x(p,C),\\ y=\varphi(p)x(p,C)+\psi(p)\end{cases}$ |

of the integral curves.

## Mathematics Subject Classification

34A05*no label found*

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