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Unformatted text preview: 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 ) 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 grating factor μ that depends only on x : μ = μ ( x ) . Then ( IF ) reads μ ( 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...
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 Spring '08
 Turner
 Exponential Function, Trigraph

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