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Lesson 9a - Integrating Factors

Lesson 9a - Integrating Factors - NonExactEquations f xy x...

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Non Exact Equations We call an equation non exact if     ) , ( , , ) , ( y x f y x Q y x P y x f yx x y xy If we try to solve non exact equations using our potential function method, we will hit a snag (when solving for c(x) or c(y), these will not be single variable functions, they will be multivariable functions. So we cannot solve for them!!) This means that we must find an integrating factor I(x,y) that when multiplied by our initial ordinary differential equation we will have our new P(x,y) d Q( y satisfy the exact property and Q(x,y) satisfy the exact property. In general: This is a tough thing to do! So we will be working on two different pes of asier integrating factors to try If they fail we will not try to do types of easier integrating factors to try. If they fail, we will not try to do any other Integrating factors.
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Integrating Factor Consider the ODE as , and consider an integrating factor I(x,y) that will make this exact. That is: 0 ' ) , ( ) , ( y y x Q y x P This means that 0 ' ) , ( ) , ( ) , ( ) , ( y y x Q y x I y x P y x I )) , ( ) , ( ( )) , ( ) , ( ( y x Q y x I x y x P y x I y If we use the product rule we get: As we can see this is already turning messy. However, assume that our ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( y x Q y x I y x Q y x I y x P y x I y x P y x I x x y y
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Lesson 9a - Integrating Factors - NonExactEquations f xy x...

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