Differential equation with total differential

An ordinary differential equation

(1)

whose left-hand side is a total derivative:

In other words, equation (1) is a differential equation with total differential if there exists a differentiable function such that

identically with respect to all arguments. The solution of a differential equation with total differential of order is reduced to solving an equation of order :

Let be an times continuously-differentiable function and let be a function having continuous partial derivatives up to and including the second order. Let

For equation (1) to be a differential equation with total differential it is sufficient that the functions , , are independent of and that [1]. In particular, may enter in a linear manner only.

The first-order equation

(2)

where the functions , , , and are defined and continuous in an open simply-connected domain of the -plane and in , is a differential equation with total differential if and only if

The general solution of equation (2) with total differential has the form , where

and the integral is taken over any rectifiable curve lying inside and joining an arbitrary fixed point with the point [2]. Equation (2) (in the general case, an equation (1) which is linear with respect to ) can, under certain conditions, be reduced to a differential equation with total differential by multiplying by an integrating factor.

References

[1]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1947)
[2]	N.P. Erugin, "A general course in differential equations" , Minsk (1972) (In Russian)