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IntegratingFactors

# IntegratingFactors - since y 4xy 3y 2" x = 4x 6y x x 2...

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Integrating Factors Given the O.D.E. M(x,y) dx + N(x,y) dy = 0 if ! M(x,y) ! y " ! N(x,y) ! x , then the equation is NOT exact or non-exact. In this case we can multiply the non-exact equation by some expression p(x,y) that will transform the original equation into an exact equation. Definition : If the O.D.E. M(x,y) dx + N(x,y) dy = 0 is non-exact in a domain D, but the equation p(x,y) M(x,y) dx + p(x,y) N(x,y) dy = 0 is an exact equation in D, then the new equation is called essentially equivalent exact equation and expression p(x,y) is called an integrating factor of the equation. Example : 1) Given the O.D.E. y dx + 2x dy = 0 since ! M ! y = 1 and ! N ! x = 2 , then it is non-exact. Consider the expression p(x,y) = y and multiply the equation by p(x,y), y 2 dx + 2xy dy = 0 this new equation is exact. So it is the essential equivalent exact equation and p(x,y) = y is an integrating factor of the equation. 2) Given the O.D.E. (4xy + 3y 2 – x) dx + x(x + 2y) dy = 0
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Unformatted text preview: since ! ! y 4xy + 3y 2 " x ( ) = 4x + 6y # ! ! x x 2 + 2xy ( ) = 2x + 2y , it is non-exact If we multiply the equation by the expression p(x,y) = x 2 , we get x 2 (4xy + 3y 2 – x) dx + x 3 (x + 2y) dy = 0 then ! ! y 4x 3 y + 3x 2 y 2 " x 3 ( ) = 4x 3 + 6x 2 y = ! ! x x 4 + 2x 3 y ( ) this new equation is exact. So, it is the essential equivalent exact equation and p(x,y) = x 2 is an integrating factor. Remark : The one-parameter family of solutions of the essential equivalent exact equation is also a solution of the non-exact equation. However, the multiplication by the integrating factor may result in either: a) The lost of (one or more) solutions of the non-exact equation. b) The gain of (one or more) solutions of the non-exact equation. c) both cases. Later on we will see examples of these case and moreover that it is not always possible to find an integrating factor....
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