Inexact differential equation

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

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

satisfying the condition

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

Leonhard Euler invented the integrating factor in 1739 to solve these equations.

Solution method

To solve an inexact differential equation, it may be transformed into an exact differential equation by finding an integrating factor $\mu$. 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:

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

Expanding this condition gives:

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

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

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

or

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

See Also

• Inexact differential
• Exact differential equation

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