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Is there an algorithm to find the integrating factor for any given first order differential equation or can we only solve it by guesswork or trial and error method? If not what should I look for while considering any first order non-exact differential equation?

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I found the following which hopefully will help you. It is from Lamar University:

http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx

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  • $\begingroup$ Sorry but this page explains only for some special forms of non-exact ODEs. I'm looking for a more general method to find the I.F. in a non-exact ODE. $\endgroup$ – Siddharth Joshi Mar 6 '16 at 5:59
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For the non-linear, non-exact first order O.D.E $f(x,y)=M(x,y) + N(x,y)y'=0$, the integrating factor $\mu(x,y)$ can be written as $$\mu(x,y) = e^{\int{\frac{M_y - N_x}{N}dx}}$$. Multiplying $f(x,y)$ by $\mu(x,y)$ will make $f$ exact.

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