Inexact differential equation explained

An inexact differential equation is a differential equation of the form (see also: inexact differential)

M(x,y) dx + N(x,y) dy =0 , where (\partial M)/(\partial y) \ne (\partial N)/(\partial x) .

The solution to such equations came with the invention of the integrating factor by Leonhard Euler in 1739.^[1]

Solution method

\mu to multiply the equation by. We'll start with the equation itself. Mdx + Ndy =0 , so we get \mu Mdx + \mu Ndy =0 . We will require \mu to satisfy $\backslash frac=\backslash frac$. We get(\partial\mu)/(M + \partial y) (\partial M)/(\mu = \partial y) (\partial\mu)/(N + \partial x) (\partial N)/(\partial x) \mu. After simplifying we getM\mu _{y-N\mu} _{x+(M} _{y-N} _{x)\mu} = 0 . Since this is a partial differential equation, it is mostly extremely hard to solve, however in some cases we will get either \mu (x,y) = \mu (x) or \mu (x,y) = \mu (y) , in which case we only need to find \mu with a first-order linear differential equation or a separable differential equation, and as such either\mu(y) = e ^{- \int{ (M _{y-N} _{x})/(dy M)} }or\mu(x) = e ^{\int{ (M _{y-N} _{x})/(dx N)} }.

Further reading

• Book: Tenenbaum, Morris . Harry . Pollard . Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences . New York . Dover . 1963 . 80–91 . 0-486-64940-7 . Recognizable Exact Differential Equations . https://books.google.com/books?id=iU4zDAAAQBAJ&pg=PA80 . registration .

External links

• A solution for an inexact differential equation from Stack Exchange
• a guide for non-partial inexact differential equations at SOS math

Notes and References

1. Web site: History of differential equations – Hmolpedia. www.eoht.info. 2016-10-16.