How to find integrating factors

Sometimes integrating factors are so simple that they can be detected by simple inspection. 

We have already established, in 26, that an integrating factor always exists which will make the equation Mdx + Ndy — 0, 

The right side of this equation is a complete differential, conse- 

Examples.—(1) Show that pt y - xif)-1 is an integrating factor of [aPy + ffldz - 2xy2dy = 0. 

Examples.—(1) Solve (x2 + y2)dx - 2xydy = 0. Ansr. x2 - y = Cx. Hint. Show f(x) = -2ar-i. The integrating factor is e- -i -2 = a-2. Prove that this is an integrating factor, and solve as in the preceding section. 

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