1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Differential Equation Integral.

  1. Oct 27, 2008 #1
    1. The problem statement, all variables and given/known data.
    {{f(x)}, {{{f}^{\prime}}{\left(x\right)}}, {{{f}^{\prime\prime}}{\left(x\right)}},...,} = {{f}, {{f}^{\prime}}, {{f}^{\prime\prime}},...,}

    Prove (without just differentiating the RHS),
    {{\int_{}^{}}{{f}^{\prime}}{\left({{f}+{{f}^{\prime\prime}}}\right)}{dx}} = { {{\frac{1}{2}}\left({f}^{2}}+{{{f}^{\prime}}^{2}\right)}+{C} }

    2. Relevant equations.
    Knowledge of Calculus and Differential Equations.

    3. The attempt at a solution.
    In the lecture notes the above problem was presented as part of another proof. I'm really not sure where to begin on this. Maybe integration by parts?


    EDIT: Thanks Mark44 for the edit.
    Last edited: Oct 28, 2008
  2. jcsd
  3. Oct 27, 2008 #2


    User Avatar
    Homework Helper

    Expand out the integrand, then split it into two integrals.

    For example

    [tex]\int x(x+1) dx = \int (x^2+x)dx= \int x^2 dx + \int x dx[/tex]
  4. Oct 27, 2008 #3


    Staff: Mentor

    You're missing some factors on the RHS. It should be 1/2 [f']^2 + 1/2 [f'']^2 + C.

    You don't need integration by parts; an ordinary substitution will do for each integral. integration with
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook