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2.6 Exact Equations - GE 207K Exact Equations September 8...

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GE 207K Exact Equations September 8, 2011 Summary: A di ff erential equation which can be written in the form M ( x, y ) + N ( x, y ) y = 0 . is exact IF M y ( x, y ) = N x ( x, y ) Once a di ff erential equation has been identified as exact, then there exists a function ψ such that it satisfies the following two conditions: ψ x = M ( x, y ) , ψ y = N ( x, y ) . The following steps can be taken to find the general solution to the di ff erential equation: Steps: 1. Write the given di ff erential equation in the STANDARD FORM M ( x, y ) + N ( x, y ) y = 0 . 2. Check for exactness: If M y = N x exact. (1) If not, not exact. 3. Recall that ψ x = M and ψ y = N . Either integrate M with respect to x OR integrate N with respect to y to find ψ . Choose which ever is easier to integrate. So: ψ = M dx + h ( y ) (2) OR ψ = N dy + h ( x ) (3) Remember to add an arbitrary function (not a constant) after integrating as shown above. ! 4. To find the arbitary function h , di ff erentiate ψ you obtained and equating it with the corresponding function M or N : If you chose ψ = M dx + h ( y ) then di ff erentiate ψ with respect to y and equate the resulting expression to N . Find what h ( y ) is by comparing the two sides. If you chose ψ = N dy + h ( x ) then di ff erentiate ψ with respect to x and equate the resulting expression to M . Find what h ( x ) is by comparing the two sides. 5. Solution is ψ = C . (4) 1 c hf, 2011
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GE 207K Exact Equations September 8, 2011 Examples solved in class: Example 1 (1 y sin x ) dx + (cos x ) dy = 0 Step 1: Find functions M and N and check for exactness. M = 1 y sin x M y = sin x N = cos x N x = sin x M y = N x and therefore the equation is exact. Step 2: Find function ψ . Note that we can integrate either M or N with ease. So let’s choose to integrate M with respect to x : ψ x = M ψ = M dx = (1 y sin x dx ) dx = x + y cos x + h ( y ) (5) where h ( y ) is a result of integrating the multivariable function ψ with respect to x .
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