A. AlacaMATH 1005Winter 20108Integrating factor for non-exact diﬀerential equationsIt is sometimes possible to convert a non-exact DE into an exact DE by multiplyingit an integrating factorI(x, y):P(x, y)+Q(x, y)y±=0(*)(non-exact.)I(x, y)P(x, yI(x, y)Q(x, y)y±= 0(exact)∂∂y(IP)=∂∂x(IQ)orIyP+y=IxQ+xorPIy-QIx=I(Qx-Py)(**)In general, to solve (**) is harder than to solve (*).But ifI=I(x) (orI=I(y)), then we can solve (**).IfI=I(x), thenIy= 0 and (**) becomes-QIx=(Qx-Py)I=⇒dIdx=Py-QxQI=⇒dII=Py-QxQdx.After we integrate and rise both side toe, we obtain|I|=e±1Q(Py-Qx)dx.
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This note was uploaded on 03/08/2010 for the course MATH 1005 taught by Professor Any during the Winter '07 term at Carleton CA.