Calculus II - 13
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This document discusses numerical methods for solving differential equations. It introduces direction fields, which provide a graphical approach to studying solutions, and Euler's method, which provides a numerical approach. Euler's method works by approximating the slope of the tangent line at each step using small step sizes to iteratively calculate successive approximations of the solution.
Calculus II - 13
- 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. = +
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- 13. Direction fields. = + , ( )= , , , ,
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- 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 = + , ( )=
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- 29. Euler’s method = + , ( )= = .
- 30. Euler’s method = + , ( )= = .