Elementary Differential Equations 8th edition by Boyce ch04

Elementary Differential Equations, with ODE Architect CD

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—————————————————————————— —— CHAPTER 4. ________________________________________________________________________ page 146 Chapter Four Section 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function 1> œ> ab , are continuous everywhere . Hence solutions are valid on the entire real line. 3. Writing the equation in standard form, the coefficients are functions with rational singularities at and . Hence the solutions are valid on the intervals , , >œ! >œ" _! a b !" "_ , , and , . 4. The coefficients are continuous everywhere, but the function is defined 1> œ68> and continuous only on the interval , . Hence solutions are defined for positive reals. !_ 5. Writing the equation in standard form, the coefficients are functions with a rational singularity at . Furthermore, is , and hence not Bœ" :Bœ> + 8B ÎB" !% a b undefined continuous, at , Hence solutions are defined on any B œ„ #5" Î# 5 œ!ß"ß#ßâÞ 5 1 interval does not that contain or . BB !5 6. Writing the equation in standard form, the coefficients are functions with rational singularities at . Hence the solutions are valid on the intervals , , Bœ„# _ # a b # # # _ , , and , . 7. Evaluating the Wronskian of the three functions, . Hence the [ 0 ß0 ß0 œ "% "#$ functions are linearly . independent 9. Evaluating the Wronskian of the four functions, . Hence the [ 0 ß0 ß0 ß0 œ! "#$% functions are linearly . To find a linear relation among the functions, we need dependent to find constants , not all zero, such that -ß-ß-ß- -0 > -0 > -0 > -0 > œ! "" ## $$ %% . Collecting the common terms, we obtain a b - #- - >  #- - - > $- - - œ! # $% "$% "#% # , which results in equations in unknowns. Arbitrarily setting , we can three four -œ" % solve the equations , , , to find that , -  #- œ " #-  - œ "  $-  - œ " - œ #Î( # $ "$ "# " - œ "$Î( - œ  $Î( #$ , . Hence #0 >  "$0 >  $0 >  (0 > œ ! " #$% . 10. Evaluating the Wronskian of the three functions, . Hence the [ 0 ß0 ß0 œ"&' functions are linearly . independent
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—————————————————————————— —— CHAPTER 4. ________________________________________________________________________ page 147 11. Substitution verifies that the functions are solutions of the ODE. Furthermore, we have [ "ß -9= >ß =38 > œ " ab . 12. Substitution verifies that the functions are solutions of the ODE. Furthermore, we have . [ "ß>ß-9=>ß=38> œ" 14. Substitution verifies that the functions are solutions of the ODE. Furthermore, we have . [ "ß>ß/ ß>/ œ/ > > #> 15. Substitution verifies that the functions are solutions of the ODE. Furthermore, we have . [ "ßBßB œ'B $ 16. Substitution verifies that the functions are solutions of the ODE. Furthermore, we have . [ Bß B ß "ÎB œ 'ÎB # 18. The operation of taking a derivative is linear, and hence -C -C œ-C -C "" ## " # "# 5 55 . It follows that P-C -C œ-C -C : -C -C â: -C -C Þ cd ‘ " # " " # 8 "" " # 88 8 "8 " a b a b Rearranging the terms, we obtain Since and P-C -C œ-PC -PC Þ C C c d " " # # " # are solutions, . The rest follows by induction.
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Elementary Differential Equations 8th edition by Boyce ch04...

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