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