Results 1 to 5 of 5

Math Help - Differential equation

  1. #1
    Member roshanhero's Avatar
    Joined
    Aug 2008
    Posts
    180

    Differential equation

    How to solve the equation-
    (x^2+y^2)dx+2xydy=0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by roshanhero View Post
    How to solve the equation-
    (x^2+y^2)dx+2xydy=0
    I'd write it as \frac{dy}{dx} = -\frac{1}{2} \left(\frac{x^2 + y^2}{xy} \right) = -\frac{1}{2} \left(\frac{x}{y} + \frac{y}{x}\right).

    Now make the usual substitution \frac{y}{x} = v \Rightarrow y = xv.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Peritus's Avatar
    Joined
    Nov 2007
    Posts
    397
    you can easily see that this is an exact ODE

    \frac{\partial }<br />
{{\partial y}}\left( {x^2  + y^2 } \right) = \frac{\partial }<br />
{{\partial x}}2xy<br />

    thus we can find the solution as follows:

    <br />
\psi (x,y) = \int {2xy} dy = xy^2  + f(x)

    we know that:

    <br />
\begin{gathered}<br />
  \frac{{\partial \psi (x,y)}}<br />
{{\partial x}} = y^2  + f'(x) = x^2  + y^2  \hfill \\<br />
   \Leftrightarrow f(x) = \frac{{x^3 }}<br />
{3} + k \hfill \\ <br />
\end{gathered}

    thus the implicit solution is:


    \frac{{x^3 }}<br />
{3} + xy^2  = c
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member roshanhero's Avatar
    Joined
    Aug 2008
    Posts
    180
    Quote Originally Posted by mr fantastic View Post
    I'd write it as \frac{dy}{dx} = -\frac{1}{2} \left(\frac{x^2 + y^2}{xy} \right) = -\frac{1}{2} \left(\frac{x}{y} + \frac{y}{x}\right).

    Now make the usual substitution \frac{y}{x} = v \Rightarrow y = xv.
    Thanks-
    I subsituted v=y/x and got the expression-
    2v/1+3v^2 dv=-dx/x.
    I tried to integrate the left side with respect to v, but I am just completely lost in this case.I wasnot able to integrate it
    So please suggest me the stage after this.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by roshanhero View Post
    Thanks-
    I subsituted v=y/x and got the expression-
    2v/1+3v^2 dv=-dx/x.
    I tried to integrate the left side with respect to v, but I am just completely lost in this case.I wasnot able to integrate it
    So please suggest me the stage after this.
    \frac{dy}{dx} = -\frac{1}{2} \left(\frac{x}{y} + \frac{y}{x}\right) \Rightarrow x \frac{dv}{dx} = \frac{-3v^2 - 1}{2v}

    (the algebra leading to this should be routine at this level. You do realise that if y = vx then dy/dx is v + x dv/dx, right?).

    This DE is seperable: \frac{dx}{x} = \frac{-2v \, dv}{3v^2 + 1}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Partial Differential Equation satisfy corresponding equation
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: May 16th 2011, 07:15 PM
  2. Replies: 4
    Last Post: May 8th 2011, 12:27 PM
  3. Replies: 1
    Last Post: April 11th 2011, 01:17 AM
  4. Partial differential equation-wave equation(2)
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: September 6th 2009, 08:54 AM
  5. Partial differential equation-wave equation - dimensional analysis
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: August 28th 2009, 11:39 AM

Search Tags


/mathhelpforum @mathhelpforum