Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

bettyboop8904

For what values of r does the function y=e^(rx) satisfy the equation y"+6y'+8y=0?

  • one year ago
  • one year ago

  • This Question is Closed
  1. dumbcow
    Best Response
    You've already chosen the best response.
    Medals 1

    just solve the quadratic formed by characteristic equation: r^2 +6r +8 = 0

    • one year ago
  2. bettyboop8904
    Best Response
    You've already chosen the best response.
    Medals 0

    the function \[y=e ^{rx}\] satisfy the equation \[y"+6y'+8y=0\]

    • one year ago
  3. bettyboop8904
    Best Response
    You've already chosen the best response.
    Medals 0

    can you elaborate a little more = (

    • one year ago
  4. dumbcow
    Best Response
    You've already chosen the best response.
    Medals 1

    no but i will refer to a site that can :) http://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx

    • one year ago
  5. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    this is a linear second order homogeneous differential equation that involves the first second and 0th derivatives of a function. Therefore the solution to this differential equation will have a solution of the form e^(rx) where r is uniquely determined by the coefficients of the 1st 2nd and 0th derivatives of y.

    • one year ago
  6. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    by using the quadratic formula like dumbcow advised, you can find the values of r which will serve as the coefficients of the linearly independent solutions of the form e^rx

    • one year ago
  7. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    Solving differential equations is all about classifying them by their order and type, and then solving accordingly by pre-prescribed methods

    • one year ago
  8. bettyboop8904
    Best Response
    You've already chosen the best response.
    Medals 0

    so you plug r in for y's? how come?

    • one year ago
  9. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    yes, sort of. you look at the equation of this form. ay"+by′+cy=0, and you take the coefficients a, b, and c and then plug them into a corresponding 2nd degree polynomial in terms of r. so for ay"+by′+cy=0, you have \[ar^2+br+c =0\] solving this polynomial will give you the values of r for which you will have solutions of the form \[e ^{rx}\]

    • one year ago
  10. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    so, lets say you obtained two real values of r 2 and 3. Therefore you will have a solution of the form \[Ke ^{3x}+Ce ^{2x}\] where C and K are arbitrary constants

    • one year ago
  11. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    each value of r gives you a linearly independent solution to the differential equation, by taking a linear combination of the two you obtain all possible solutions

    • one year ago
  12. bettyboop8904
    Best Response
    You've already chosen the best response.
    Medals 0

    ok so i understand the first part the solving this polynomial part is confusing and so we're using the quadratic formula with a=1 b=6 and c=8?

    • one year ago
  13. dumbcow
    Best Response
    You've already chosen the best response.
    Medals 1

    yes, however you should notice that it can be easily factored as well

    • one year ago
  14. bettyboop8904
    Best Response
    You've already chosen the best response.
    Medals 0

    I'm sorry you are helping me but your examples are a little fancy worded for me. I'm really good with actual problems but when math starts adding describing words I start getting confused lol = (

    • one year ago
  15. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    my b, lol

    • one year ago
  16. bettyboop8904
    Best Response
    You've already chosen the best response.
    Medals 0

    its ok lol

    • one year ago
  17. bettyboop8904
    Best Response
    You've already chosen the best response.
    Medals 0

    so if you factor it then what would you do with the values?

    • one year ago
  18. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    so yea, set up your 2nd degree polynomial using the coefficients from the initial equation: y"+6y'+8y=0 \[r^2 +6r+8=0\] like dumbcow said, the left side of this equation can be easily factored: \[(r+4)(r+2)=0\] the solutions to this equation are therefore r=-4 and r=-2 so therefore we have two linearly independent solutions \[y = e ^{-4x}\] and \[y = e ^{-2x}\] both of these equations are solutions to the differential equation y"+6y'+8y=0. Every possible linear combination of these equations is as well, ad since they are linearly independent, the sum of these equations is also a solution to y"+6y'+8y=0. Try all of these out use \[y = e ^{-4x}\], \[y = e ^{-2x}\], \[y = 4e ^{-4x}\] and \[y = e ^{-4x}+e ^{-2x}\]

    • one year ago
  19. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    you'll find that they will all satisfy y"+6y'+8y=0. So in order to provide a general equation that expresses all possible solutions we write \[y=Ke ^{-4x}+Ce ^{-2x}\] where K and C are arbitrary constants

    • one year ago
  20. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    and that's the answer

    • one year ago
  21. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    plug in any value for K and C and then plug that into y"+6y'+8y and you will get 0

    • one year ago
  22. bettyboop8904
    Best Response
    You've already chosen the best response.
    Medals 0

    ok i get it now = ) thank you for the help = )

    • one year ago
  23. sweet1137
    Best Response
    You've already chosen the best response.
    Medals 0

    np

    • one year ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.