An inexact differential equation is a differential equation of the form: :

M(x,y) \, dx+N(x,y) \, dy=0

satisfying the condition :

\frac \ne \frac

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

invented the

integrating factor In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve non-exact ordinary differential equations, but is also used within multivari ...

in 1739 to solve these equations.

Solution method

To solve an inexact differential equation, it may be transformed into an

exact differential equation In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering. Definition Given a simply connected and open subset ''D'' of \mat ...

by finding an

\mu

. Multiplying the original equation by the integrating factor gives: :

\mu M\,dx+\mu N\,dy=0

. For this equation to be exact,

\mu

must satisfy the condition: :

\frac=\frac

. Expanding this condition gives: :

M\mu_y-N\mu_x+(M_y-N_x)\mu = 0.

Since this is a

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

, it is generally difficult. However in some cases where

\mu

depends only on

x

y

, the problem reduces to a separable

first-order linear differential equation In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) where and are arbi ...

. The solutions for such cases are: :

\mu(y)=e^

or :

\mu(x)=e^.

References

External links

A solution for an inexact differential equation
from Stack Exchange
a guide for non-partial inexact differential equations
at SOS math {{Differential equations topics, state=collapsed Equations Ordinary differential equations Differential calculus Discrete mathematics Mathematical structures

Solution method

See Also

References

Further reading

External links