General investigations of the partial differential equations of the first order. Different forms of the integrability conditions

Abstract

We shall now concern ourselves with the general investigations of first order partial differential equations. We assume that the function to be found does not itself appear in the differential equation. This assumption is not an essential restriction since the general case can always be reduced to this. In fact, if the given differential equation contains the function V to be found has the form

$$0 = \phi \left( {V,\frac{{\partial V}} {{\partial {q_1}}},\frac{{\partial V}} {{\partial {q_2}}}, \ldots ,\frac{{\partial V}} {{\partial {q_n}}},{q_1},{q_2}, \ldots ,{q_n}} \right)$$

, then one introduces a new independent variable q and a new dependent variables W through the equation

$$W = qV$$

; then

$$\frac{{\partial W}} {{\partial q}} = V,\frac{{\partial W}} {{\partial {q_1}}} = q\frac{{\partial V}} {{\partial {q_1}}}, \ldots ,\frac{{\partial W}} {{\partial {q_n}}} = q\frac{{\partial V}} {{\partial {q_n}}}$$

, so

$$V = \frac{{\partial W}} {{\partial q}},\frac{{\partial V}} {{\partial {q_1}}} = \frac{1} {q}\frac{{\partial W}} {{\partial {q_1}}}, \ldots ,\frac{{\partial V}} {{\partial {q_n}}} = \frac{1} {q}\frac{{\partial W}} {{\partial {q_n}}}$$

. Therefore the given differential equation goes over into the following:

$$0 = \phi \left( {\frac{{\partial W}} {{\partial q}},\frac{1} {q}\frac{{\partial W}} {{\partial {q_1}}}, \ldots ,\frac{1} {q}\frac{{\partial W}} {{\partial {q_n}}},{q_1},{q_2}, \ldots ,{q_n}} \right)$$

, which indeed contains one more independent variable q, in which, however, W itself does not occur, but only its differential coefficients with respect to q₁, q₂, …, q _n , q. We can therefore confine ourselves, without limiting the generality,to the case where

$$\varphi \left( {\frac{{\partial V}} {{\partial {q_1}}},\frac{{\partial V}} {{\partial {q_2}}}, \ldots ,\frac{{\partial V}} {{\partial {q_n}}},{q_1},{q_2}, \ldots ,{q_n}} \right) = 0$$

is the given differential equation and V itself does not occur in the equation.