Elementary Differential Equations 8th edition by Boyce ch08

Elementary Differential Equations, with ODE Architect CD

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
—————————————————————————— —— CHAPTER 8. ________________________________________________________________________ page 438 Chapter Eight Section 8.1 2. The Euler formula for this problem is C 2 & > $ C 8" 8 8 8 ˆ‰ È , in which Since , we can also write >8 2 Þ >œ! 8 !! C œ C  &82  $2 C 8" 8 8 # È , with . Cœ# ! ab + . Euler method with 2 œ !Þ!& À 8œ# 8œ% 8œ' 8œ) > !Þ" !Þ# !Þ$ !Þ% C "Þ&**)! "Þ#*#)) "Þ!(#%# !Þ*$!"(& 8 8 , 2 œ !Þ!#& À . Euler method with 8œ"# 8œ"' > !Þ" !Þ# !Þ$ !Þ% C "Þ'""#% "Þ$"$'" "Þ"!!"# !Þ*'#&&# 8 8 The Euler formula is backward C 2 & > $ C 8" 8 8 8 È +1 +1 , in which Since , we can also write 2 Þ 8 C œ C  & 8  " 2  $2 C 8" 8 8" # È , with . Solving for , and choosing the C ! 8" positive root, we find that C œ  2 #!8#* 2 %C Þ $" ## 8" 8 # # ”• È
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
—————————————————————————— —— CHAPTER 8. ________________________________________________________________________ page 439 ab - . Backward Euler method with 2 œ !Þ!& À 8œ# 8œ% 8œ' 8œ) > !Þ" !Þ# !Þ$ !Þ% C "Þ'%$$( "Þ$("'% "Þ"(('$ "Þ!&$$% 8 8 . 2 œ !Þ!#& À . Backward Euler method with 8œ"# 8œ"' > !Þ" !Þ# !Þ$ !Þ% C "Þ'$$!" "Þ$&#*& "Þ"&#'( "Þ!#%!( 8 8 3. The Euler formula for this problem is C œC2# C$ > 8" 8 8 8 , in which Since , >8 2 Þ >œ! 8 !! C œ C  #2C  $82 8" 8 8 # , with . Cœ" ! + . Euler method with 2 œ !Þ!& À > !Þ" !Þ# !Þ$ !Þ% C "Þ#!#& "Þ%"'!$ "Þ'%#)* "Þ))&*! 8 8 , 2 œ !Þ!#& À . Euler method with > !Þ" !Þ# !Þ$ !Þ% C "Þ#!$)) "Þ%"*$' "Þ'%)*' "Þ)*&(# 8 8 The Euler formula is backward C 2 # C $ > 8" 8 8 8" +1 , in which Since , we can also write 2 Þ 8 C œ C  #2 C  $ 8  " 2 8" 8 8" # , with . Solving for , C ! 8" we find that Þ C$8"2 "#2 8" 8 #
Background image of page 2
—————————————————————————— —— CHAPTER 8. ________________________________________________________________________ page 440 ab - . Backward Euler method with 2 œ !Þ!& À 8œ# 8œ% 8œ' 8œ) > !Þ" !Þ# !Þ$ !Þ% C "Þ#!)'% "Þ%$"!% "Þ'(!%# "Þ*$!(' 8 8 . 2 œ !Þ!#& À . Backward Euler method with 8œ"# 8œ"' > !Þ" !Þ# !Þ$ !Þ% C "Þ#!'*$ "Þ%#')$ "Þ''#'& "Þ*")!# 8 8 4. The Euler formula is C 2 # > / B : > C Þ 8" 8 8 8 8 cd Since and , we can also write >8 2 >œ! 8 !! C œC #82 2/B: 82C 8" 8 8 # , with Cœ" Þ ! + . Euler method with 2 œ !Þ!& À > !Þ" !Þ# !Þ$ !Þ% C "Þ"!#%% "Þ#"%#' "Þ$$%)% "Þ%'$** 8 8 , 2 œ !Þ!#& À . Euler method with > !Þ" !Þ# !Þ$ !Þ% C "Þ"!$'& "Þ#"'&' "Þ$$)"( "Þ%')$# 8 8 The Euler formula is backward C 2 # > / B : >C Þ 8" 8 8" 8" 8" Since and , we can also write > œ8"2 ! 8" C œC #2 8" 2/B:  8" 2C 8" 8 8" # c d , with This equation cannot be solved for . At each step, given Þ C ! explicitly 8" the current value of , the equation must be solved C 8 numerically for 8"
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
—————————————————————————— —— CHAPTER 8. ________________________________________________________________________ page 441 ab - . Backward Euler method with 2 œ !Þ!& À 8œ# 8œ% 8œ' 8œ) > !Þ" !Þ# !Þ$ !Þ% C "Þ"!(#! "Þ##$$$ "Þ$%(*( "Þ%)""! 8 8 . 2 œ !Þ!#& À . Backward Euler method with 8œ"# 8œ"' > !Þ" !Þ# !Þ$ !Þ% C "Þ"!'!$ "Þ##""! "Þ$%%($ "Þ%(')) 8 8 6. The Euler formula for this problem is C œ C  2 >  C =38 C 8" 8 8 88 ## ˆ‰ . Here and . So that >œ! >œ8 2 ! 8 C œ C  2 8 2  C =38 C 8" 8 8 # 8 , with .
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 01/24/2008.

Page1 / 60

Elementary Differential Equations 8th edition by Boyce ch08...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online