Inexact differential equation

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

${\displaystyle M(x,y)\,dx+N(x,y)\,dy=0}$

satisfying the condition

${\displaystyle {\frac {\partial M}{\partial y}}\neq {\frac {\partial N}{\partial x}}}$

Leonhard Euler invented the integrating factor in 1739 to solve these equations. ^[1]

Solution method

To solve an inexact differential equation, it may be transformed into an exact differential equation by finding an integrating factor ${\displaystyle \mu }$. ^[2] Multiplying the original equation by the integrating factor gives:

${\displaystyle \mu M\,dx+\mu N\,dy=0}$.

For this equation to be exact, ${\displaystyle \mu }$ must satisfy the condition:

${\textstyle {\frac {\partial \mu M}{\partial y}}={\frac {\partial \mu N}{\partial x}}}$.

Expanding this condition gives:

${\displaystyle M\mu _{y}-N\mu _{x}+(M_{y}-N_{x})\mu =0.}$

Since this is a partial differential equation, it is generally difficult. However in some cases where ${\displaystyle \mu }$ depends only on ${\displaystyle x}$ or ${\displaystyle y}$, the problem reduces to a separable first-order linear differential equation. The solutions for such cases are:

${\displaystyle \mu (y)=e^{\int {{\frac {N_{x}-M_{y}}{M}}\,dy}}}$

or

${\displaystyle \mu (x)=e^{\int {{\frac {M_{y}-N_{x}}{N}}\,dx}}.}$

See Also

• Inexact differential
• Exact differential equation

References

1. "History of differential equations – Hmolpedia". www.eoht.info. Retrieved 2016-10-16.
2. "Special Integrating Factors" (PDF). people.clas.ufl.edu. Retrieved 2025-02-08.

Further reading

• Tenenbaum, Morris; Pollard, Harry (1963). "Recognizable Exact Differential Equations". Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences . New York: Dover. pp.  80–91. ISBN   0-486-64940-7.

External links

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