1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Calculus of Variations - more confusion

  1. Jan 22, 2009 #1
    I asked a question earlier about Calculus of Variation, but the question I gave didn't really highlight my confusion well. I've come across some other questions that I think reveal my misunderstanding.

    1. The problem statement, all variables and given/known data

    Solve the Euler equation for the following integral:

    (integral from x1->x2) ∫[(y')² + √y]dx

    2. Relevant equations

    Euler equation:

    ∂F/∂y - d/dx (∂F/∂y') = 0

    3. The attempt at a solution

    F = (y')² + √y

    So ∂F/∂y = 1/[2√y] and d/dx (∂F/∂y') = 2y'

    Thus: y'' = 1/[4√y]

    Although it has been some time since I took an ODE course, I think that the equation above is non-trivial to solve. So either I'm mistaken and this is easy to solve or I'm going about the calculus of variations method mistakenly.

    With a similar problem, ∫[1+yy']²dx, I was only able to reduce it down to: y''y² + y(y')² = 0, which again I couldn't solve.

    I'm not looking for answers, just want to know where I'm applying the method incorrectly, or if in fact I'm missing a far easier way to apply it.

    Thanks for any help!
     
  2. jcsd
  3. Jan 22, 2009 #2
    Any thoughts?

    Is y'' = 1/[4√y] easily solvable? If not how else can the method be applied?

    Cheers
     
  4. Jan 22, 2009 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    As far as calculus of variations goes, they look fine. Just because you get a difficult ODE doesn't mean the method is wrong.
     
  5. Jan 22, 2009 #4

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Try solving for y' first by noting that

    [tex]y''(x)=\frac{d}{dx}y'(x)=\left(\frac{d}{dy}\frac{dy}{dx}\right)y'(x)=\frac{dy'}{dy}y'[/tex]
     
    Last edited: Jan 22, 2009
  6. Jan 22, 2009 #5
    Thanks gabbagabbahey, that was a big help.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Calculus of Variations - more confusion
  1. Calculus of variations (Replies: 8)

  2. Calculus of variations (Replies: 0)

  3. Calculus by variations (Replies: 2)

  4. Variational Calculus (Replies: 0)

  5. Calculus of variations (Replies: 3)

Loading...