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[parent] d'Alembert's equation (Derivation)

The first order differential equation

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

$\displaystyle y = \varphi(p)\cdot x+\psi(p).$
We take $ p$ as a new variable and derive the equation with respect to $ p$, getting
$\displaystyle p-\varphi(p) = [x\varphi'(p)+\psi'(p)]\frac{dp}{dx}.$
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
$\displaystyle y = p_{\nu}x+\psi(p_{\nu}) \quad (\nu = 1, 2, ..., k)$
for the original equation. If $ \varphi(p) \not\equiv p$, then the derived equation may be written as
$\displaystyle \frac{dx}{dp} = \frac{\varphi'(p)}{p-\varphi(p)}x+\frac{\psi'(p)}{p-\varphi(p)},$
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 representation
\begin{displaymath}\begin{cases} x = x(p, C),\ y = \varphi(p)x(p, C)+\psi(p) \end{cases}\end{displaymath}
of the integral curves.



"d'Alembert's equation" is owned by pahio.
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See Also: Clairaut's equation, contraharmonic proportion

Other names:  Lagrange equation

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Cross-references: integral curves, general solution, linear differential equation, solutions, roots, variable, equation, real functions, differentiable, differential equation
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This is version 13 of d'Alembert's equation, born on 2004-08-02, modified 2005-06-13.
Object id is 6058, canonical name is DAlembertsEquation.
Accessed 4350 times total.

Classification:
AMS MSC: 34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions)
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