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Homework Help: Using substitution in differential eq

  1. Aug 15, 2010 #1
    xdy/dx+y=1/y^2:using substitution in differential eq

    1. The problem statement, all variables and given/known data
    solve using substitution
    xdy/dx+y=1/y^2


    3. The attempt at a solution
    Thanks to the people who've help me thus far. here's a bernulli problem that I'm having. I change this problem around to...
    dy/dx=y^3/xy^2
    xy^2dy=y^3dx

    using u sub.
    u=y^3
    du=3y^2dy

    substituted problem

    1/3xdu=udx
    du/dx=3xu
    du/dx-3xu=0

    then I get e^(integral -3x)=e^(-3x^2/2)

    Here's where I'm stuck

    e^(-3x^2/2)u=integral 0*e^(-3x^2/2)

    doesn't that just have c? which later becomes
    u=ce^(3x^2/2)
    However, that's not the solution of the equation which is
    y^3=1+cx^-3

    Can someone explain why?
     
    Last edited: Aug 15, 2010
  2. jcsd
  3. Aug 15, 2010 #2

    LCKurtz

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    Science Advisor
    Homework Helper
    Gold Member

    There are three terms in your original DE, so you have dropped one. Also you need to be careful whether your x is in the numerator or denominator as you work. Use parentheses when there is doubt. After your substitution your DE should look like this:

    [tex]\frac 1 3 x u' + u = 1[/tex]

    Once you put that in correct form and find the integrating factor, you shouldn't have any ex terms.
     
  4. Aug 16, 2010 #3

    ehild

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    Homework Helper

    And what about substituting u=xy?

    ehild
     
  5. Aug 16, 2010 #4

    Mark44

    Staff: Mentor

    The DE is separable, which is something you should check for at the start in problems like this.

    The original equation is equivalent to
    [tex]\frac{y^2 dy}{1 - y^3} = \frac{dx}{x}[/tex]
     
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