Elementary Differential Equations 8th edition by Boyce ch03

Elementary Differential Equations, with ODE Architect CD

This preview shows pages 1–3. Sign up to view the full content.

—————————————————————————— —— CHAPTER 3. ________________________________________________________________________ page 83 Chapter Three Section 3.1 1. Let , so that and . Direct substitution into the differential C œ / C œ < / C œ < / <> w <> ww <> equation yields . Canceling the exponential, the characteristic a b <  #<  \$ / œ ! # <> equation is The roots of the equation are , . Hence the <  #<  \$ œ ! Þ < œ  \$ " # general solution is . C œ - /  - / " # > \$> 2. Let . Substitution of the assumed solution results in the characteristic equation C œ / <> <  \$<  # œ ! Þ < œ  #  " # The roots of the equation are , . Hence the general solution is . C œ - /  - / " # > #> 4. Substitution of the assumed solution results in the characteristic equation C œ / <> #<  \$<  " œ ! Þ < œ "Î# " # The roots of the equation are , . Hence the general solution is . C œ - /  - / " # >Î# > 6. The characteristic equation is , with roots . Therefore the %<  * œ ! < œ „\$Î# # general solution is . C œ - /  - / " # \$>Î# \$>Î# 8. The characteristic equation is , with roots . Hence the <  #<  # œ ! < œ "„ \$ # È general solution is . C œ - /B: "  \$ >  - /B: "  \$ > " # Š Š È È 9. Substitution of the assumed solution results in the characteristic equation C œ / <> <  <  # œ ! Þ < œ  # " # The roots of the equation are , . Hence the general solution is . Its derivative is . Based on the C œ - /  - / C œ  #- /  - / " # " # #> > w #> > first condition, , we require that . In order to satisfy , C ! œ " -  - œ " C ! œ " a b a b " # w we find that . Solving for the constants, and . Hence the  #-  - œ " - œ ! - œ " " # " # specific solution is . C > œ / a b > 11. Substitution of the assumed solution results in the characteristic equation C œ / <> '<  &<  " œ ! Þ < œ "Î\$ "Î# # The roots of the equation are , . Hence the general solution is . Its derivative is . Based C œ - /  - / C œ - / Î\$  - / Î# " # " # >Î\$ >Î# w >Î\$ >Î# on the first condition, , we require that . In order to satisfy the C ! œ " -  - œ % a b " # condition , we find that . Solving for the constants, C ! œ " - Î\$  - Î# œ ! - œ "# w a b " # " and . Hence the specific solution is . - œ  ) C > œ "# /  ) / # a b >Î\$ >Î# 12. The characteristic equation is , with roots , . Therefore the <  \$< œ ! < œ  \$ ! # general solution is , with derivative . In order to C œ -  - / C œ  \$ - / " # # \$> w \$> satisfy the initial conditions, we find that , and . Hence the -  - œ  #  \$ - œ \$ " # # specific solution is . C > œ  "  / a b \$> 13. The characteristic equation is , with roots <  &<  \$ œ ! #

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

View Full Document
—————————————————————————— —— CHAPTER 3. ________________________________________________________________________ page 84 < œ  & "\$ # # "ß# È . The general solution is , with C œ - /B:  &  "\$ >Î#  - /B:  &  "\$ >Î# " # Š Š È È derivative C œ - /B:  &  "\$ >Î#  - /B:  &  "\$ >Î#  &  "\$  &  "\$ # # w È È Š Š È È " # . In order to satisfy the initial conditions, we require that , and -  - œ " " # & "\$ & "\$ # # È È -  - œ ! - œ "  &Î "\$ Î# " # " . Solving for the coefficients, and Š È - œ "  &Î "\$ Î# Þ # Š È 14. The characteristic equation is , with roots #<  <  % œ ! # < œ  " \$\$ % % "ß# È .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern