D'Alembert's differential equation

The d'alembert differential equation , also called the Lagrangian differential equation , is a nonlinear ordinary differential equation of first order of form

${\displaystyle \ y(x)=xg(y'(x))+f(y'(x)).}$

It is named after Jean-Baptiste le Rond d'Alembert . A special case of this differential equation is the Clairaut differential equation .

Procedure for finding some solutions

Let be a solution of the linear differential equation ${\displaystyle u\colon (a,b)\rightarrow \mathbb {R} }$

${\displaystyle \ [x-g(x)]u'(x)-g'(x)u(x)=f'(x)}$

and on injective with differentiable inverse function . Then ${\displaystyle u}$ ${\displaystyle (a,b)}$ ${\displaystyle u^{-1}}$

${\displaystyle y(x):=x\cdot g(u^{-1}(x))+f(u^{-1}(x))}$

a solution of the d'alembert differential equation.

proof

The following applies:

${\displaystyle {\begin{array}{lll}y'(x)&=&g(u^{-1}(x))+{\frac {xg'(u^{-1}(x))+f'(u^{-1}(x))}{u'(u^{-1}(x))}}\\&=&g(u^{-1}(x))+{\frac {xg'(u^{-1}(x))+[u^{-1}(x)-g(u^{-1}(x))]u'(u^{-1}(x))-xg'(u^{-1}(x))}{u'(u^{-1}(x))}}\\&=&u^{-1}(x)\ .\end{array}}}$

${\displaystyle \Box }$

Note, however, that in general not all solutions can be found in this way, as can be seen in the special case of Clairaut's differential equation. There one would only find the solutions called nontrivial solutions with this method.

literature

• Wolfgang Walter: Ordinary differential equations. 7th edition, Springer Verlag, Berlin 2000, ISBN 3-540-67642-2 , § 4.