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I have an equation in the form :

$ M(x,y) + N(x,y)\frac{dy}{dx} $ I know the integrating factor for this equation. My problem is I can not rearrange my equation in to the form : $ \frac{dy}{dx} + p(x)y = Q(x) $ Do I need to rearrange if the integrating factor is known?

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    Just to add, the integrating factor is defined as follows: \\[ \rho(x) = e^{\int \! P(x)\,\mathrm{d}x} \\]2011-06-01

2 Answers 2

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If you have an integrating factor $\mu(x,y)$ such that $\mu(x,y)M(x,y)dx + \mu(x,y)N(x,y)dy = 0$ is exact, then you can follow the normal procedure for exact equations to solve it. There's a completely different notion of integrating factor for solving 1st order linear equations ${dy \over dx} + P(x)y = Q(x)$; it just happens to be called the same thing since in both cases you multiply through by a factor and eventually integrate something when you're solving the differential equation.

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It actually depends on the form of M and N. But basically, you can use any technique other than getting the integrating factor to solve for the ODE, like determining whether such ODE is exact, homogeneous, etc.