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Non-Homogeneous Equations

Non-Homogeneous Equations. Method of Undetermined Coefficients. We Know How To Solve Homogeneous Equations. (With Constant Coefficients). Find Roots of Characteristic Polynomial. Determine Appropriate General Solution. But what about Non-Homogeneous Equations?.

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Non-Homogeneous Equations

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  1. Non-Homogeneous Equations • Method of Undetermined Coefficients

  2. We Know How To SolveHomogeneous Equations (With Constant Coefficients) Find Roots of Characteristic Polynomial Determine Appropriate General Solution

  3. But what about Non-Homogeneous Equations? Recall that we assumed the solution For the homogeneous equation

  4. But what about Non-Homogeneous Equations? Use as a guide For the Non-homogeneous equation, guess a different form of solution.

  5. Example

  6. Example Use to guess form of a solution suggests that (This is the undetermined coefficient)

  7. Example Use to guess form of a solution suggests that Then:

  8. Example suggests that Then: Plugging In:

  9. Example suggests that Then: Plugging In:

  10. Example suggests that Then: Plugging In:

  11. Example suggests that Then: Plugging In: These are the same

  12. Example suggests that Then: Plugging In: Specific Solution:

  13. Method of Undetermined Coefficients Use as a guide Guess that specific solution takes the form: (This is the undetermined coefficient)

  14. Method of Undetermined Coefficients Use as a guide Guess that specific solution takes the form: Plug in to differential equation Solve for

  15. Method of Undetermined Coefficients Guess that specific solution takes the form: Plug in to differential equation Solve for Determining the right Depends on (Will go through important cases later)

  16. General Solutions Undetermined Coefficients Gives one Specific Solution But Adding or Multiplying By a Constant Breaks the Solution!

  17. General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in:

  18. General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in:

  19. General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in:

  20. General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in:

  21. General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in: These are the same!

  22. General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in: No help for finding General Solutions!

  23. General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you multiply by a constant And substitute in (exercise - try it): No help for finding General Solutions!

  24. General Solutions So how do we find general solutions? Go back to the homogeneous case Find general solution, i.e. where (The “h” is for “homogeneous”)

  25. General Solutions For If is a specificsolution to the non-homogeneous equation And is the generalsolution to the homogeneous equation Then Is a general solution to the homogeneous equation

  26. General Solutions (Specific Solution) (General Homogeneous Solution) Plug in

  27. General Solutions (Specific Solution) (General Homogeneous Solution) Plug in

  28. General Solutions (Specific Solution) (General Homogeneous Solution) Plug in

  29. General Solutions (Specific Solution) (General Homogeneous Solution) Plug in

  30. General Solutions (Specific Solution) (General Homogeneous Solution) Plug in

  31. General Solutions (Specific Solution) (General Homogeneous Solution) Plug in So it is a (General) Solution

  32. Example Specific Solution: Homogeneous Equation Has General Solution (I assume you can determine this) So the Non-Homogeneous Equation Has General Solution

  33. So to solve…

  34. So to solve… Use Undetermined Coefficients to find a specific solution Find the general solution To the Homogeneous Equation

  35. So to solve… Use Undetermined Coefficients to find a specific solution Find the general solution To the Homogeneous Equation The General Solution takes the form:

  36. Summary • Method of Undetermined Coefficients Gives a Specific Solution For Non-Homogenous Equations • General Solution comes from General Solution of Homogeneous Equation • We will discuss Undetermined Coefficients More Next..

  37. Questions?

  38. Undetermined CoefficientGuesses (“Ansatz”) Form or or Times Corresponding Form Times anything above

  39. Divide and Conquer If Can Find Specific Solutions And Their Sum Will Be A Specific Solution To (The Logic Is Identical To Why Is A General Solution)

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