# d’Alembert’s equation

The first differential equation^{}

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

is called d’Alembert’s differential equation; here $\phi $ and $\psi $ some known differentiable^{} real functions.

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

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

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

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

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

$$y={p}_{\nu}x+\psi ({p}_{\nu})\mathit{\hspace{1em}}(\nu =1,2,\mathrm{\dots},k)$$ |

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

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

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

$$\{\begin{array}{cc}x=x(p,C),\hfill & \\ y=\phi (p)x(p,C)+\psi (p)\hfill & \end{array}$$ |

of the integral curves.

Title | d’Alembert’s equation |
---|---|

Canonical name | DAlembertsEquation |

Date of creation | 2013-03-22 14:31:05 |

Last modified on | 2013-03-22 14:31:05 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 16 |

Author | pahio (2872) |

Entry type | Derivation |

Classification | msc 34A05 |

Synonym | Lagrange equation^{} |

Related topic | ClairautsEquation |

Related topic | ContraharmonicProportion |

Related topic | DerivativeAsParameterForSolvingDifferentialEquations |