d'Alembert's equationDAlembertsEquation2004-08-02 17:58:072005-06-13 08:38:17Derivationdifferential equation% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe first \PMlinkescapetext{order} differential equation
$$y = \varphi(\frac{dy}{dx})\cdot x+\psi(\frac{dy}{dx})$$
is called {\em d'Alembert's differential equation}; here $\varphi$ and $\psi$ \PMlinkescapetext{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
$$p-\varphi(p) = [x\varphi'(p)+\psi'(p)]\frac{dp}{dx}.$$
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
$$\frac{dx}{dp} = \frac{\varphi'(p)}{p-\varphi(p)}x+\frac{\psi'(p)}{p-\varphi(p)},$$
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 \PMlinkescapetext{representation}
\[\begin{cases}
x = x(p, C),\\
y = \varphi(p)x(p, C)+\psi(p)
\end{cases}\]
of the integral curves.