Stewart Calculus Section 9.2

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. Direction fields.
= +

3. Direction fields.
= +

4. Direction fields.
= + , ( )=

5. Direction fields.
= + , ( )=

6. Direction fields.
= + , ( )=

7. Direction fields.
= + , ( )=

8. Direction fields.
= + , ( )=

9. Direction fields.
= + , ( )=

10. Direction fields.
= + , ( )=

11. Direction fields.
= +

12. Direction fields.
= +

13. Direction fields.
= + , ( )= , , , ,

14. Direction fields.
= + , ( )= , , , ,

15. Direction fields.
= + , ( )= , , , ,

16. Direction fields.
= + , ( )= , , , ,

17. Euler’s method
= ( , ), ( )=

18. Euler’s method
= ( , ), ( )=
Let be a small step size.

19. Euler’s method
= ( , ), ( )=
Let be a small step size.
= +
= +
= +
······

20. Euler’s method
= ( , ), ( )=
Let be a small step size.
= +
= +
= +
······
= + · ( , )
= + · ( , )
= + · ( , )
······

21. Euler’s method
= + , ( )=
Let be a small step size.
= +
= +
= +
······
= + · ( , )
= + · ( , )
= + · ( , )
······

22. Euler’s method
= + , ( )=
Let be a small step size. = .
= +
= +
= +
······
= + · ( , )
= + · ( , )
= + · ( , )
······

23. Euler’s method
= + , ( )=
Let be a small step size. = .
= + = + = .
= + = + = .
= + = + = .
······ ······
= + · ( , )
= + · ( , )
= + · ( , )
······

24. Euler’s method
= + , ( )=
Let be a small step size. = .
= + = + = .
= + = + = .
= + = + = .
······ ······
= + · ( , ) = + ·( + )= .
= + · ( , ) = + ·( + )= .
= + · ( , ) = + ·( + )= .
······ ······

25. Euler’s method
= + , ( )=

26. Euler’s method
= + , ( )=

27. Euler’s method
= + , ( )=
=

28. Euler’s method
= + , ( )=
= .

29. Euler’s method
= + , ( )=
= .

30. Euler’s method
= + , ( )=
= .