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Math Help - [SOLVED] Please solve this

  1. #1
    Member Altair's Avatar
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    [SOLVED] Please solve this

    (3x+y)dy = (x+3y)dx , y(0) = 1

    As it is a homogeneous differential equation I substituted y=vx and I can't solve it further than the following expression :-

    \frac{x}{dx} = \frac {(1+2v)}{(3+v)dv}

    Please simplify it for me.
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  2. #2
    Flow Master
    mr fantastic's Avatar
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    Quote Originally Posted by Altair View Post
    (3x+y)dy = (x+3y)dx , y(0) = 1

    As it is a homogeneous differential equation I substituted y=vx and I can't solve it further than the following expression :-

    \frac{x}{dx} = \frac {(1+2v)}{(3+v)dv}

    Please simplify it for me.
    I'm sorry but what working out you've done is wrong.

    \frac{dy}{dx} = \frac{x + 3y}{3x + y} = \frac{1 + 3 \, (\frac{y}{x})}{3 + \, (\frac{y}{x})}.

    Substitute y = xv:

     v + x \frac{dv}{dx} = \frac{1 + 3v}{3 + v} \Rightarrow x \frac{dv}{dx} = \frac{1 + 3v}{3 + v} - v = \frac{1 - v^2}{3 + v}.

    The DE is now seperable:

    \frac{3 + v}{v^2 - 1} \, dv = \frac{1}{x} \, dx

    etc.
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  3. #3
    Member Altair's Avatar
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    Oh sorry. Thanks.
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