Integrating factor for a non-exact equation

Builds on Exact equation Test for exactness Integrating factor

Summary

Let $M\, dx + N\, dy = 0$ be an equation that is not exact on an open set $D$ . An integrating factor for it is a nowhere zero differentiable function $\mu$ on $D$ such that $\mu M\, dx + \mu N\, dy = 0$ is exact. On a rectangle, with $M$ , $N$ and $\mu$ of class $C^1$ , the test for exactness makes this condition $(\mu M)_y = (\mu N)_x$ . The two equations have the same solutions, because $\mu$ has no zero.

Remarks

If $(M_y - N_x)/N$ is a function $g(x)$ of $x$ only, then $\mu(x) = e^{\int g(x)\, dx}$ is an integrating factor. If $(N_x - M_y)/M$ is a function $h(y)$ of $y$ only, then $\mu(y) = e^{\int h(y)\, dy}$ is an integrating factor. The integrating factor of a linear first-order equation is the first case: $y' + p y - q = 0$ has $M = p y - q$ , $N = 1$ and $(M_y - N_x)/N = p$ .

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Differential equation concept
Function concept
Integrating factor concept
Interval concept
Ordinary differential equation definition
Solution of an ordinary differential equation definition
Exact equation definition
Test for exactness statement
Integrating factor for a non-exact equation concept

Tag	`0CX86CR8JV`
URI	`m2://0CX86CR8JV`
Kind	concept
Writing	W0 stub
MSC	34A05
Specializes	Integrating factor
Uses	Exact equation, Test for exactness, Integrating factor

Integrating factor for a non-exact equation

Summary

Remarks

Reading path

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