Lagrange equation

An ordinary first-order differential equation, not solved for the derivative, but linear in the independent variable and the unknown function:

$$ \tag{1 } F ( y ^ \prime ) x + G ( y ^ \prime ) y = H ( y ^ \prime ) . $$

This equation is connected with the name of J.L. Lagrange (1759, see [1]); it was also investigated by J. d'Alembert, hence it is sometimes called d'Alembert's equation. A special case of the Lagrange equation is the Clairaut equation.

Lagrange's equation is always solvable in quadratures by the method of parameter introduction (the method of differentiation). Suppose, for example, that (1) can be reduced to the form

$$ \tag{2 } y = f ( y ^ \prime ) x + g ( y ^ \prime ) ,\ \ f ( y ^ \prime ) \not\equiv y ^ \prime . $$

After introducing the parameter $ p = y ^ \prime $ and taking the total differential of both sides of (2) (cf. also Total derivative), taking into account the relation $ dy = p dx $, one arrives at the first-order linear equation

$$ [ p - f ( p) ] \frac{dx}{dp} - f ^ { \prime } ( p) x = g ^ \prime ( p) . $$

If $ x = \Phi ( p , C ) $ is a solution of this equation (where $ C $ is an arbitrary constant), then the solution of (2) can be written in parametric form,

$$ x = \Phi ( p , C ) ,\ \ y = f ( p) \Phi ( p , C ) + g ( p) . $$

If $ p _ {0} $ is an isolated root of the equation $ p = f ( p) $, then $ y = f ( p _ {0} ) x + g ( p _ {0} ) $ is also a solution of (2); this solution can be singular (cf. Singular solution).

References

[1]	J.L. Lagrange, "Sur l'intégration d'une équation différentielle" J.A. Serret (ed.) , Oeuvres , 1 , G. Olms, reprint (1973) pp. 21–36
[2]	W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)

References

[a1]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)