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 œ C  2 & >  \$ C 8" 8 8 8 ˆ È , in which Since , we can also write > œ >  82 Þ > œ ! 8 ! ! C œ C  &82  \$2 C 8" 8 8 # È , with . C œ # ! a b + . Euler method with 2 œ !Þ!& À 8 œ # 8 œ % 8 œ ' 8 œ ) > !Þ" !Þ# !Þ\$ !Þ% C "Þ&**)! "Þ#*#)) "Þ!(#%# !Þ*\$!"(& 8 8 a b , 2 œ !Þ!#& À . Euler method with 8 œ % 8 œ ) 8 œ "# 8 œ "' > !Þ" !Þ# !Þ\$ !Þ% C "Þ'""#% "Þ\$"\$'" "Þ"!!"# !Þ*'#&&# 8 8 The Euler formula is backward C œ C  2 & >  \$ C 8" 8 8 8 ˆ È +1 +1 , in which Since , we can also write > œ >  82 Þ > œ ! 8 ! ! C œ C  & 8  " 2  \$2 C 8" 8 8" # a b È , with . Solving for , and choosing the C œ # C ! 8" positive root, we find that C œ 2  #!8  #* 2  %C Þ \$ " # # 8" 8 # # È a b

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

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—————————————————————————— —— CHAPTER 8. ________________________________________________________________________ page 441 a b - . Backward Euler method with 2 œ !Þ!& À 8 œ # 8 œ % 8 œ ' 8 œ ) > !Þ" !Þ# !Þ\$ !Þ% C "Þ"!(#! "Þ##\$\$\$ "Þ\$%(*( "Þ%)""!
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