02b_ExactDiff_IntegratingFactor

02b_ExactDiff_IntegratingFactor - Exact Differentials and...

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1 Exact Differentials and Integrating Factors J. R. Rice, AM 105b, 6 February 2008 Consider a first order ode which is given in the form dy dx = ± M ( x , y ) N ( x , y ) . To solve such an ode is equivalent, after multiplying by dx , and rearranging, to finding functions y = y ( x ) satisfying the differential form M ( x , y ) dx + N ( x , y ) dy = 0 As a special case. Let us suppose that the form M ( x , y ) dx + N ( x , y ) dy is a perfect differential (sometimes called an exact differential). That means that a function f ( x , y ) exists so that M ( x , y ) dx + N ( x , y ) dy = df ( x , y ) , where df ( x , y ) = ± f ( x , y ) ± x dx + ± f ( x , y ) ± y dy . Thus, if such an f ( x , y ) does exists, which means that Mdx + Ndy = df for arbitrary ratios of dx to dy, then it must be the case that M ( x , y ) = ± f ( x , y ) ± x and N ( x , y ) = ± f ( x , y ) ± y , implying that ± M ( x , y ) ± y = ± 2 f ( x , y ) ± y ± x = ± 2 f ( x , y ) ± x ± y = ± N ( x , y ) ± x To turn that around, if M and N satisfy ± M ( x , y ) ± y = ± N ( x , y ) ± x , then it can be shown (please show it) that the differential form M ( x , y ) dx + N ( x , y ) dy is perfect, i.e., that a function f ( x , y ) exists (and is unique to within an additive constant) such that M ( x , y ) dx + N ( x , y ) dy = df ( x , y ) . In such case it is trivial to solve dy dx = ± M ( x , y ) N ( x , y ) , and the solutions are just f ( x , y ) = C (an arbitrarily chosen constant). If we want the specific solution satisfying y = b when x = a , we just choose the constant C = f ( a , b ) . Of course, in general if we are given two functions M ( x , y ) and N ( x , y ) , it must be anticipated that ± M ( x , y ) ± y ² ± N ( x , y ) ± x , and hence that M ( x , y ) dx + N ( x , y ) dy is not a perfect
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2 differential. Thus the above approach fails. But let us be optimists and ask as follows: Noting that the original ode could have equivalently been written as dy dx = ± ² ( x , y ) M ( x , y ) ( x , y ) N ( x
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This note was uploaded on 02/29/2008 for the course MATH 105b taught by Professor Rice during the Spring '08 term at Harvard.

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02b_ExactDiff_IntegratingFactor - Exact Differentials and...

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