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Calculus II - 13

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Stewart Calculus Section 9.2

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Calculus II - 13

  1. 1. 9.2 Direction Fields and Euler’s Method Most differential equation cannot be solved explicitly. We can study the solution through a graphical approach (direction fields) or a numerical approach (Euler’s method).
  2. 2. Direction fields. = +
  3. 3. Direction fields. = +
  4. 4. Direction fields. = + , ( )=
  5. 5. Direction fields. = + , ( )=
  6. 6. Direction fields. = + , ( )=
  7. 7. Direction fields. = + , ( )=
  8. 8. Direction fields. = + , ( )=
  9. 9. Direction fields. = + , ( )=
  10. 10. Direction fields. = + , ( )=
  11. 11. Direction fields. = +
  12. 12. Direction fields. = +
  13. 13. Direction fields. = + , ( )= , , , ,
  14. 14. Direction fields. = + , ( )= , , , ,
  15. 15. Direction fields. = + , ( )= , , , ,
  16. 16. Direction fields. = + , ( )= , , , ,
  17. 17. Euler’s method = ( , ), ( )=
  18. 18. Euler’s method = ( , ), ( )= Let be a small step size.
  19. 19. Euler’s method = ( , ), ( )= Let be a small step size. = + = + = + ······
  20. 20. Euler’s method = ( , ), ( )= Let be a small step size. = + = + = + ······ = + · ( , ) = + · ( , ) = + · ( , ) ······
  21. 21. Euler’s method = + , ( )= Let be a small step size. = + = + = + ······ = + · ( , ) = + · ( , ) = + · ( , ) ······
  22. 22. Euler’s method = + , ( )= Let be a small step size. = . = + = + = + ······ = + · ( , ) = + · ( , ) = + · ( , ) ······
  23. 23. Euler’s method = + , ( )= Let be a small step size. = . = + = + = . = + = + = . = + = + = . ······ ······ = + · ( , ) = + · ( , ) = + · ( , ) ······
  24. 24. Euler’s method = + , ( )= Let be a small step size. = . = + = + = . = + = + = . = + = + = . ······ ······ = + · ( , ) = + ·( + )= . = + · ( , ) = + ·( + )= . = + · ( , ) = + ·( + )= . ······ ······
  25. 25. Euler’s method = + , ( )=
  26. 26. Euler’s method = + , ( )=
  27. 27. Euler’s method = + , ( )= =
  28. 28. Euler’s method = + , ( )= = .
  29. 29. Euler’s method = + , ( )= = .
  30. 30. Euler’s method = + , ( )= = .

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