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L05 - Integrating Factors Suppose the FOODE M(x y dx N(x y...

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Integrating Factors Suppose the FOODE M ( x, y ) dx + N ( x, y ) dy = 0 is not exact. The previous example leads to the question: Can we multiply it by some function μ = μ ( x, y ) , to be called an Integrating factor , so that ( μ ( x, y ) M ( x, y ) ) dx + ( μ ( x, y ) N ( x, y ) ) dy = 0 is exact? We need μ y M + μM y = μ x N + μN x . ( IF )
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Integrating Factors Suppose the FOODE M ( x, y ) dx + N ( x, y ) dy = 0 is not exact. The previous example leads to the question: Can we multiply it by some function μ = μ ( x, y ) , to be called an Integrating factor , so that ( μ ( x, y ) M ( x, y ) ) dx + ( μ ( x, y ) N ( x, y ) ) dy = 0 is exact? We need μ y M + μM y = μ x N + μN x . ( IF ) This is a PDE for μ , much too hard. Sometimes it it possible to find an inte-
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grating factor μ that depends only on x : μ = μ ( x ) . Then ( IF ) reads μ 0 ( x ) = μ ( x ) M y ( x, y ) - N x ( x, y ) N ( x, y ) . ( IF x ) If the quotient on the right does not de- pend on y , we have our μ = μ ( x ) : it is the exponential of the indefinite x –integral of this quotient. Sometimes it it possible to find an inte- grating factor μ that depends only on y : μ = μ ( y ) . Then ( IF ) reads μ 0 ( y ) = μ ( y ) N
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