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Exact Equations

Exact Equations. Nonlinear Non-separable First Order Equations. Techniques We Know. Nonlinear Separable. Separation of Variables. Integrating Factors. Linear. Non-separable. Nonlinear. Implicit Differentiation. Note:. Implicit Differentiation. Implicit Differentiation.

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Exact Equations

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  1. Exact Equations • Nonlinear Non-separable First Order Equations

  2. Techniques We Know Nonlinear Separable Separation of Variables Integrating Factors Linear Non-separable Nonlinear

  3. Implicit Differentiation Note:

  4. Implicit Differentiation

  5. Implicit Differentiation This is the general rule: This comes from the terms with x This comes from the terms with y

  6. Implicit Differentiation This is the general rule: So to solve problems of the form: What do we do? Find: Where:

  7. Solving Nonlinear Non-Separable Equations To solve problems of the form: Find: Where: We’ll talk about how to do this in a minute

  8. Solving Nonlinear Non-Separable Equations To solve problems of the form: Find: Where: Rewrite as: Integrate:

  9. Solving Nonlinear Non-Separable Equations To solve problems of the form: Find: Where: Rewrite as: Integrate:

  10. Finding f(x,y) Something Something (maybe different)

  11. Finding f(x,y) Nothing in this case

  12. Finding f(x,y)

  13. The catch… (Because there’s always a catch) Try to find: Where: (Hint: You’re going to be trying for a really long time…) (Hint: Because it can’t be done…)

  14. Exact Equations How do we know? Remember, order doesn’t matter for partial derivatives: So, if: Then:

  15. Exact Equations How do we know? Remember, order doesn’t matter for partial derivatives: So, if: Then: These are the same!

  16. Exact Equations How do we know? Remember, order doesn’t matter for partial derivatives: So, if: Then:

  17. Exact Equations For Equations of the form: If: (and we know how to solve) The equation is called Exact Example:

  18. Exact Equations For Equations of the form: If: (and we know how to solve) The equation is called Exact Not an Example:

  19. Solving Nonlinear Non-Separable Equations To solve problems of the form: See if Equation is Exact: If it is Exact, Find: Where: Rewrite as: Integrate:

  20. Example

  21. Example Equation is Exact!

  22. Example Equation is Exact!

  23. Example Equation is Exact! General Solution

  24. Summary Non-separable Nonlinear Exact Method of Exact Equations

  25. Summary Separation of Variables Nonlinear Separable Integrating Factors Linear Non-separable Nonlinear Exact Method of Exact Equations

  26. Questions?

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