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Substitution methods

  1. Apr 17, 2005 #1
    Hi, I'm new to the forum, and new to differential equations. I was wondering if someone could post a no-nonsence explanation of substitution methods for first order differential equations.

    Thanks!
     
  2. jcsd
  3. Apr 17, 2005 #2

    dextercioby

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    I don't know what u may be speaking about.Are u referring to a separable diff.eq.?

    [tex] a(x)\frac{dy}{dx}+b(x)f(y)=0 [/tex]

    Daniel.
     
  4. Apr 17, 2005 #3
    Sorry if my post was confusing... I meant first order differential equations that are neither linear nor separable.
     
  5. Apr 17, 2005 #4

    dextercioby

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    U mean something like that?

    [atex] a(x)\left(\frac{dy}{dx}\right)^{k}+b(x)f(y)=c(x) [/tex]

    The homegenous equation is separable.Therefore integrable.

    Daniel.
     
  6. Apr 17, 2005 #5
    I mean one like this:
     

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  7. Apr 17, 2005 #6

    dextercioby

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    The homogenous equation is separable.

    Daniel.
     
  8. Apr 17, 2005 #7
    I guess that's what I don't understand. How is it separable? I've been working on this for an hour, and I can't separate the variables. Could you show me how to work it out?

    I really want to understand this, because I have to teach the concept of substitution methods to the rest of my class in a few weeks. My book says that this equation is not separable, and that the homogenous equation is only separable via substitution. And thats all it says.
     
    Last edited: Apr 17, 2005
  9. Apr 17, 2005 #8

    dextercioby

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    The homogenous equation is

    [tex] 2xy\frac{dy}{dx}=3y^{2} [/tex]

    Since [itex] y\neq 0 [/itex]

    ,u get

    [tex] 2x\frac{dy}{dx}=3y [/tex]

    ,separate variables & integrate to get

    [tex] y_{hom}(x)=Cx^{3/2} [/tex]

    Now,apply Lagrange's method to find the particular solution of the nonhomogenous one.

    Daniel.
     
  10. Apr 17, 2005 #9
    I'm sorry, what's Lagrange's method??? Its not in my book. I don't think its supposed to be that complicated of a solution, this is chapter 1 ODE stuff. There is no way to solve this problem by substitution?
     
  11. Apr 17, 2005 #10

    dextercioby

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    Yes it can,make the substitution

    [tex] y^{2}(x)=u(x) [/tex]

    Daniel.
     
  12. Apr 17, 2005 #11
    What happened to 4x^2?
     
  13. Apr 17, 2005 #12

    dextercioby

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    It's there in the RHS,that substitution simplifies the integration of the ODE...



    Daniel.
     
  14. Apr 17, 2005 #13

    saltydog

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    That equation is homogeneous of degree 2. Thus, you can make a substitution y=vx, and then separate variables. Your ODE book should have this technique as a separate section at the beginning unless you have one of those "qualitative books" like Devaney's. I'm old-school.
     
  15. Apr 17, 2005 #14
    Thanks! Thats the substitution I came up with, I must have made a mistake after that point.
     
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