Quantcast

A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

johnny0929

  • one year ago

Solve the initial value problem\[u''+6u'+12u=0\]with \(u(0)=1\) and \(u'(0)=0\)

  • This Question is Closed
  1. modphysnoob
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    use characteristic equation a^2 + 6a+12 solve for a

  2. abb0t
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    this is a second order homogeneous differential equation. do you know the general solution? \[y_p = e^{rt}c_1+e^{rt}c_2\]

  3. abb0t
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @modphysnoob is correct. except he used "a" so your solution would be: \[y_p = e^{at}c_1+e^{-at}c_2\]

  4. abb0t
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    solve the quadratic

  5. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I actually have the answer but I am just a little confused about how to do it... It's a spring mass problem and the answer is \[u=e^{-3t}cos\sqrt{3}t+\sqrt{3}e^{-3t}sin\sqrt{3}t\]

  6. abb0t
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    then, plug in the initial conditions. take a few derivatives and violê

  7. abb0t
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    solve for u(0)=1 take the derivative of your particular solution and plug in u(0)=0

  8. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok. when I factor \(a^2+6a+12=0\) I get \((-i a+\sqrt{3}-3 i) (i a+\sqrt{3}+3 i)\)

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

    that mean it is sinosoid

  10. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    which means it's \((\sqrt{3}\pm(3+a)i)=0\) how do i make it look like the general solution?

  11. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    can you help? because I'm so confused

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

    hold on, my computer is apparently running on steam engine

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

    a=-3 (+-) i* sqrt(3) which is \[Ae^{(-3+ i \sqrt{3})t}+Be^{(-3- i \sqrt{3})t}\]

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

    use euler identity and you will get A sin( ) + B cos ()

  15. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    now i can see it's \[Ae^{-3}sin(\sqrt{3}t)+Be^{-3}cos(\sqrt{3}t)\]then all i have to do is plug in the initial values, right?

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

    you are missing t e^(-3t)

  17. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh yeah. ok there's a t.

  18. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[u=e^{−3t}cos\sqrt{3}t+\sqrt{3}e^{−3t}sin\sqrt{3}t\]one more question. how do i determine if this equation is over damped, underdamped, or critically damped?

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

    so sin() and cos(x) just go up and down at t go on

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

    |dw:1362965561866:dw|

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

    while e(-3t) |dw:1362965602529:dw|

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

    just read it http://en.wikipedia.org/wiki/Damping#Critical_damping_.28.CE.B6_.3D_1.29 it will explain much better than I ever can

  23. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh ok i am reading from my book that all i need to do is compare the coefficient in front of \(u'\) to \(\sqrt{4ac}\) from the equation \(a^2+6a+12=0\)

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

    crticle damp goes to zero without oscillating under dampm oscillate

  25. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    thank you!

  26. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy

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.