Results 1 to 2 of 2

Math Help - Differential equations help with substitution

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    297

    Differential equations help with substitution

    Solve:

    y'=\dfrac{2}{(x+2y-3)} by letting x+2y-3=v

    I always get confused up as to what we're meant to do with the y'

    y'=\dfrac{2}{v}

    Is \dfrac{dv}{dx}=1+2\dfrac{dy}{dx}?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,547
    Thanks
    1418
    If x + 2y - 3 = v

    2y = v - x + 3

    y = \frac{1}{2}v - \frac{1}{2}x + \frac{3}{2}

    \frac{d}{dx}(y) = \frac{d}{dx}\left(\frac{1}{2}v - \frac{1}{2}x + \frac{3}{2}\right)

    \frac{dy}{dx} = \frac{1}{2}\,\frac{d}{dx}(v) - \frac{1}{2}\,\frac{d}{dx}(x) + \frac{d}{dx}\left(\frac{3}{2}\right)

    \frac{dy}{dx} = \frac{1}{2}\,\frac{d}{dv}(v)\,\frac{dv}{dx} - \frac{1}{2}

    \frac{dy}{dx} = \frac{1}{2}\,\frac{dv}{dx} - \frac{1}{2}.


    So substituting into your DE

    \frac{dy}{dx} = \frac{2}{x + 2y - 3}

    \frac{1}{2}\,\frac{dv}{dx} - \frac{1}{2} = \frac{2}{v}

    \frac{dv}{dx} - 1 = \frac{4}{v}

    \frac{dv}{dx} = \frac{4}{v} + 1

    \frac{dv}{dx} = \frac{v + 4}{v}

    \frac{dx}{dv} = \frac{v}{v + 4}

    \frac{dx}{dv} = \frac{v + 4}{v + 4} - \frac{4}{v + 4}

    \frac{dx}{dv} = 1 - \frac{4}{v + 4}

    x = \int{1 - \frac{4}{v + 4}\,dv}

    x = v - 4\ln{|v + 4|} + C

    x = x + 2y - 3 - 4\ln{|x + 2y - 3 + 4|} + C

    x = x + 2y - 3 - 4\ln{|x + 2y + 1|} + C

    4\ln{|x+2y+1|} = 2y - 3 + C

    \ln{|x+2y+1|}= \frac{1}{2}y - \frac{3}{4} + C

    |x + 2y + 1| = e^{\frac{1}{2}y - \frac{3}{4} + C}

    |x + 2y + 1| = e^{-\frac{3}{4} + C}e^{\frac{1}{2}y}

    x + 2y + 1 = \pm e^{-\frac{3}{4} + C}e^{\frac{1}{2}y}

    x + 2y + 1 = Ae^{\frac{1}{2}y}

    x = Ae^{\frac{1}{2}y} - 2y - 1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Differential Equation Substitution
    Posted in the Differential Equations Forum
    Replies: 10
    Last Post: January 28th 2011, 09:39 AM
  2. Differential equation with substitution
    Posted in the Differential Equations Forum
    Replies: 8
    Last Post: January 5th 2011, 03:42 AM
  3. Differential equations substitution
    Posted in the Differential Equations Forum
    Replies: 8
    Last Post: September 3rd 2010, 03:00 AM
  4. Differential equations substitution
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: September 2nd 2010, 09:20 PM
  5. Replies: 2
    Last Post: May 18th 2009, 03:49 AM

Search Tags


/mathhelpforum @mathhelpforum