Integrating factor for non-exact differential equations - Copy (2)

Integrating factor for non-exact differential equations - Copy (2)

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A. Alaca MATH 1005 Winter 2010 8 Integrating factor for non-exact differential equations It is sometimes possible to convert a non-exact DE into an exact DE by multiplying it an integrating factor I ( x, y ): P ( x, y ) + Q ( x, y ) y = 0 ( * ) (non-exact.) I ( x, y ) P ( x, y ) + I ( x, y ) Q ( x, y ) y = 0 (exact) ∂y ( IP ) = ∂x ( IQ ) or I y P + IP y = I x Q + IQ x or PI y - QI x = I ( Q x - P y ) ( ** ) In general, to solve ( ** ) is harder than to solve ( * ). But if I = I ( x ) (or I = I ( y )), then we can solve ( ** ). If I = I ( x ), then I y = 0 and ( ** ) becomes - QI x = ( Q x - P y ) I = dI dx = P y - Q x Q I = dI I = P y - Q x Q dx. After we integrate and rise both side to e , we obtain
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