L05 - Integrating Factors Suppose the FOODE M ( x,y ) dx +...

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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 = ( x,y ) , to be called an Integrating factor , so that ( ( x,y ) M ( x,y ) ) dx + ( ( x,y ) N ( x,y ) ) dy = 0 is exact? We need y M + M y = x N + N 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 = ( x,y ) , to be called an Integrating factor , so that ( ( x,y ) M ( x,y ) ) dx + ( ( x,y ) N ( x,y ) ) dy = 0 is exact? We need y M + M y = x N + N x . ( IF ) This is a PDE for , much too hard. Sometimes it it possible to find an inte- grating factor that depends only on x : = ( x ) . Then ( IF ) reads ( x ) = ( 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 = ( x ) : it is the...
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L05 - Integrating Factors Suppose the FOODE M ( x,y ) dx +...

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