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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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(1) A nitpick: what you wrote in the first displayed formula is not an equation. An equation must have an equals sign somewhere in it. (2) Whether you need to rearrange depends on what you intend to do with it. If you just want to show us: oh look, here be an ODE, then no, you don't need to rearrange it. If you want to do certain computations, or perhaps solve the ODE explicitly, then rearrangement can sometimes make things easier. – Willie Wong Jan 21 '11 at 16:03
HAHAHA @Willie$\ldots$ +1 – night owl Jun 1 '11 at 16:35
Just to add, the integrating factor is defined as follows: \[ \rho(x) = e^{\int \! P(x)\,\mathrm{d}x} \] – night owl Jun 1 '11 at 16:37
up vote 1 down vote accepted

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.

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