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Homework Help: Problem solving separable diff EQ w/ I.Condition. , they say its wrong :x

  1. Jan 14, 2006 #1
    Hello everyone i did this one surley thing it would work out and yet another failure. :cry:
    Solve the separable differential equation
    11 x - 8 y*sqrt{x^2 + 1}*{dy}/{dx} = 0.
    Subject to the initial condition: y(0) = 6.
    y = ?


    I'm pretty sure where I messed up is when i tried to solve for y, i htink i screwed up there but i'm not sure, anyone know? Thanks! its #11. Ignore that top stuff please!

    Here is my work:
    http://img57.imageshack.us/img57/5705/lastscan8cm.jpg [Broken]
     
    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Jan 14, 2006 #2

    benorin

    User Avatar
    Homework Helper

    Particular solution

    You problem lies in where you placed your constant of integration, namely:

    [tex]11\int \frac{x}{\sqrt{x^2+1}}dx = \frac{11}{2}\int \frac{1}{\sqrt{u}}du = \frac{11}{2}\int u^{-\frac{1}{2}}du = 11u^{\frac{1}{2}}+C=11\sqrt{x^2+1}+C[/tex]

    C is outside the square root, from there

    [tex]4y^2=11\sqrt{x^2+1}+C[/tex]

    [tex]y^2=\frac{11}{4}\sqrt{x^2+1}+\frac{C}{4}[/tex]

    or,

    [tex]y=\pm\sqrt{\frac{11}{4}\sqrt{x^2+1}+\frac{C}{4}}[/tex]

    to solve for C, use the simplest form of the solved DE, that is use

    [tex]4y^2=11\sqrt{x^2+1}+C[/tex]

    for x=0 and y(0)=6, this gives

    [tex]4(6)^2=11\sqrt{0^2+1}+C[/tex]

    which simplifies to

    [tex]144=11+C[/tex]

    and hence C=133, plug this into

    [tex]y=\pm\sqrt{\frac{11}{4}\sqrt{x^2+1}+\frac{C}{4}}[/tex]

    to get

    [tex]y(x)=\pm\sqrt{\frac{11}{4}\sqrt{x^2+1}+\frac{133}{4}}=\pm\frac{1}{2}\sqrt{11\sqrt{x^2+1}+133}[/tex]

    as your particular solution.
     
    Last edited: Jan 14, 2006
  4. Jan 15, 2006 #3
    yup, that constant shouldn't be under the radical! :tongue:
     
  5. Jan 15, 2006 #4
    Ahhh, that worked perfectly Benorin, thanks a ton (again)! :) The step by step explanation is great!
     
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