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 > a b , 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 a b a b ! " " _ , , and , . 4. The coefficients are continuous everywhere, but the function is defined 1 > œ 68 > a b and continuous only on the interval , . Hence solutions are defined for positive reals. a b ! _ 5. Writing the equation in standard form, the coefficients are functions with a rational singularity at . Furthermore, is , and hence not B œ " : B œ >+8 BÎ B  " ! % a b a b undefined continuous, at , Hence solutions are defined on any B œ „ #5  " Î# 5 œ !ß "ß #ß â Þ 5 a b 1 interval does not that contain or . B B ! 5 6. Writing the equation in standard form, the coefficients are functions with rational singularities at . Hence the solutions are valid on the intervals , , B œ „ #  _  # a b a b a b  # # # _ , , and , . 7. Evaluating the Wronskian of the three functions, . Hence the [ 0 ß 0 ß 0 œ  "% a b " # \$ functions are linearly . independent 9. Evaluating the Wronskian of the four functions, . Hence the [ 0 ß 0 ß 0 ß 0 œ ! a b " # \$ % 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 ! " " # # \$ \$ % % a b a b a b a b . Collecting the common terms, we obtain a b a b 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 ! " # \$ % a b a b a b a b . 10. Evaluating the Wronskian of the three functions, . Hence the [ 0 ß 0 ß 0 œ "&' a b " # \$ 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 " a b . 12. Substitution verifies that the functions are solutions of the ODE. Furthermore, we have . [ "ß >ß -9= >ß =38 > œ " a b 14. Substitution verifies that the functions are solutions of the ODE. Furthermore, we have . [ "ß >ß / ß > / œ / a b > > #> 15. Substitution verifies that the functions are solutions of the ODE. Furthermore, we have . [ "ß Bß B œ 'B a b \$ 16. Substitution verifies that the functions are solutions of the ODE. Furthermore, we have . [ Bß B ß "ÎB œ 'ÎB a b # 18. The operation of taking a derivative is linear, and hence a b - C  - C œ - C  - C " " # # " # " # a b a b a b 5 5 5 . It follows that P - C  - C œ - C  - C  : - C  - C  â  : - C  - C Þ c d c d " " # # " # " " # 8 " " # # " # " # 8 8 8" 8" a b a b a b a b Rearranging the terms, we obtain Since and P - C  - C œ - P C  - P C Þ C C c d c d c d " " # # " " # # " # are solutions, . The rest follows by induction. P - C  - C œ ! c d " " # # 19 . Note that , for . a b a b a b a b + . > Î.>
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