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Differential Equation Integral.

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

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

    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?

    Thanks,

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

    rock.freak667

    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

    Mark44

    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
     
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