MATH24 Exact DE (Q1) - Consider the General Form of DE:...

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Consider the General Form of DE: M(x,y) dx + N(x,y) dy = 0 Suppose separation of variables does not hold, assume that there is a function F(x,y) such that M = and N = . If M is differentiated with respect to y and N with respect to x, that is but Therefore, which is a necessary and sufficient condition to be an exact equation. Exact differential equations may be solved using any of the four methods: 1. Integrable Combination 2. Partial Derivatives 3. Line integral 4. Alternative Solution 1. Integrable Combinations Integrable combinations consist of group of terms that forms an exact differential, thus it is readily integrable. It may be obtained by rearranging the terms in the given DE until a group of terms forms an integrable combination . Some of the integrable combinations are listed below: 1. xdy + ydx = d(xy) 2. 3. 4. 5. 6. 7. 8. 9. mx m – 1 y n dx + nx m y n – 1 dy = d(x m y n ) 10. mx m – 1 y n dx – nx m y n – 1 dy/(y n ) 2 = Examples: Test for exactness and find the general solution. 1.
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MATH24 Exact DE (Q1) - Consider the General Form of DE:...

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