Elementary Differential Equations 8th edition by Boyce ch08

# Elementary Differential Equations, with ODE Architect CD

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—————————————————————————— —— 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

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—————————————————————————— —— 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 #
—————————————————————————— —— 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"

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

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