d’Alembert’s equation

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

is called d’Alembert’s differential equation; here $\varphi$ and $\psi$ 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

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

of the integral curves.

Title d’Alembert’s equation DAlembertsEquation 2013-03-22 14:31:05 2013-03-22 14:31:05 pahio (2872) pahio (2872) 16 pahio (2872) Derivation msc 34A05 Lagrange equation  ClairautsEquation ContraharmonicProportion DerivativeAsParameterForSolvingDifferentialEquations