Differential equation with total differential

An ordinary differential equation

$$ \tag{1 } F ( x , y , y ^ \prime \dots y ^ {( n) } ) = 0 $$

whose left-hand side is a total derivative:

$$ \frac{d}{dx} \Phi ( x , y , y ^ \prime \dots y ^ {( n - 1 ) } ) = 0 . $$

In other words, equation (1) is a differential equation with total differential if there exists a differentiable function $ \Phi ( x , u _ {0} \dots u _ {n - 1 } ) $ such that

$$ F ( x , u _ {0} \dots u _ {n} ) \equiv \Phi _ {x} ^ \prime + u _ {1} \Phi _ {u _ {0} } ^ \prime + + \dots + u _ {n} \Phi _ {u _ {n-} 1 } ^ \prime $$

identically with respect to all arguments. The solution of a differential equation with total differential of order $ n $ is reduced to solving an equation of order $ ( n - 1 ) $:

$$ \Phi ( x , y , y ^ \prime \dots y ^ {( n - 1 ) } ) = C ,\ \ C = \textrm{ const } . $$

Let $ F ( x, u _ {0} \dots u _ {n} ) $ be an $ n $ times continuously-differentiable function and let $ \Phi ( x , u _ {0} \dots u _ {n - 1 } ) $ be a function having continuous partial derivatives up to and including the second order. Let

$$ \Delta \Phi = \Phi _ {x} ^ \prime + u _ {1} \Phi _ {u _ {0} } ^ \prime + \dots + u _ {n} \Phi _ {u _ {n-} 1 } ^ \prime , $$

$$ \Delta _ {0} F = F _ {u _ {n} } ^ { \prime } ,\ \Delta _ \nu F = F _ {u _ {n - \nu } } ^ { \prime } - \Delta ( \Delta _ {\nu - 1 } F ) ,\ \nu = 1 \dots n . $$

For equation (1) to be a differential equation with total differential it is sufficient that the functions $ \Delta _ \nu F $, $ \nu = 0 \dots n $, are independent of $ u _ {n} $ and that $ \Delta _ {n } F = 0 $[1]. In particular, $ u _ {n} $ may enter $ F $ in a linear manner only.

The first-order equation

$$ \tag{2 } M ( x , y ) + N ( x , y ) y ^ \prime = 0 , $$

where the functions $ M $, $ N $, $ M _ {y} ^ { \prime } $, and $ N _ {x} ^ { \prime } $ are defined and continuous in an open simply-connected domain $ D $ of the $ ( x , y ) $- plane and $ M ^ {2} + N ^ {2} > 0 $ in $ D $, is a differential equation with total differential if and only if

$$ M _ {y} ^ { \prime } ( x , y ) \equiv N _ {x} ^ { \prime } ( x , y ) \ \mathop{\rm in} D . $$

The general solution of equation (2) with total differential has the form $ \Phi ( x , y ) = 0 $, where

$$ \Phi ( x , y ) = \int\limits _ {( x _ {0} , y _ {0} ) } ^ { {( } x , y ) } M ( x , y ) dx + N ( x , y ) dy , $$

and the integral is taken over any rectifiable curve lying inside $ D $ and joining an arbitrary fixed point $ ( x _ {0} , y _ {0} ) \in D $ with the point $ ( x , y ) $[2]. Equation (2) (in the general case, an equation (1) which is linear with respect to $ y ^ {(} n) $) 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)