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Solving Exact Equations with Integrating Factors

Learn to solve exact equations, identify integrating factors, and solve IVP problems. Practice problems included. Find the largest interval over which solutions are defined.

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Solving Exact Equations with Integrating Factors

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  1. Warm-up Problems Solve the IVP . Give the largest interval over which the solution is defined.

  2. Exact Equations Recall that we defined the differential of z = f (x,y) as dz = fx dx + fy dy Ex. Consider x2 – 5xy + y3 = c, find the differential of both sides.

  3. M(x,y)dx + N(x,y)dy is an exact differential if it is the differential of some function f (x,y). A DE of the form M(x,y)dx + N(x,y)dy = 0 is called an exact equation. • But how can we identify an exact equation? Thm.M(x,y)dx + N(x,y)dy is an exact differential iff

  4. Ex. Solve 2xydx + (x2 – 1)dy = 0.

  5. Ex. Solve (e2y – y cos xy)dx + (2xe2y – x cos xy + 2y)dy = 0.

  6. Ex. Solve

  7. Practice Problems Section 2.4 Problems 2, 22

  8. To make M(x,y)dx + N(x,y)dy exact, we multiply by an integrating factor: If is in terms of only x, use If is in terms of only y, use

  9. Ex. Solve xydx + (2x2 + 3y2 – 20)dy = 0.

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