Results 1 to 2 of 2

Thread: Non Linear homogenous differential equation

  1. May 28th 2008, 04:06 AM #1
    dankelly07
    dankelly07 is offline
    Member
    Joined
    Sep 2007
    Posts
    76

    Non Linear homogenous differential equation

    Just need a bit of help finishing this one, not sure if I'm going about this correctly...

    <br /> \begin{gathered}<br /> \frac{{dy}}<br /> {{dx}} = \frac{{y^2 + 3xy + x^2 }}<br /> {{x^2 }} \hfill \\<br /> \frac{{dy}}<br /> {{dx}} = \frac{{v^2 + 3v}}<br /> {x} - \frac{v}<br /> {x} \hfill \\<br /> \frac{{dy}}<br /> {{dx}} = v^2 + 2v/x \hfill \\<br /> \hfill \\<br /> \int {\frac{{dv}}<br /> {{v(v + 2v)}} = \int {\frac{{dx}}<br /> {x} = > \int {(\frac{1}<br /> {v} - \frac{1}<br /> {{1 + v}})dv = \int {\frac{{dx}}<br /> {x}} } } } \hfill \\<br /> \end{gathered} <br />


    <br /> \ln |v/2 + v| = \ln |x| + A<br />

    ......

    I need to know how to get to this....

    <br /> \frac{x}<br /> {{A - \ln |x|}} - x<br />
    Follow Math Help Forum on Facebook and Google+
  2. May 28th 2008, 05:44 AM #2
    Isomorphism
    Isomorphism is offline
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,460
    Quote Originally Posted by dankelly07 View Post
    Just need a bit of help finishing this one, not sure if I'm going about this correctly...

    <br /> \begin{gathered}<br /> \frac{{dy}}<br /> {{dx}} = \frac{{y^2 + 3xy + x^2 }}<br /> {{x^2 }} \hfill \\<br /> \frac{{dy}}<br /> {{dx}} = \frac{{v^2 + 3v}}<br /> {x} - \frac{v}<br /> {x} \hfill \\<br /> \frac{{dy}}<br /> {{dx}} = v^2 + 2v/x \hfill \\<br /> \hfill \\<br /> \int {\frac{{dv}}<br /> {{v(v + 2v)}} = \int {\frac{{dx}}<br /> {x} = > \int {(\frac{1}<br /> {v} - \frac{1}<br /> {{1 + v}})dv = \int {\frac{{dx}}<br /> {x}} } } } \hfill \\<br /> \end{gathered} <br />


    <br /> \ln |v/2 + v| = \ln |x| + A<br />

    ......

    I need to know how to get to this....

    <br /> \frac{x}<br /> {{A - \ln |x|}} - x<br />
    \frac{dy}{dx} = \frac{y^2 + 3xy + x^2 }{x^2}

    With y = vx substitution:

    v + x\frac{dv}{dx} = v^2 + 3v + 1
    x\frac{dv}{dx} = v^2 + 2v + 1 = (v+1)^2
    \int \frac{dv}{(v+1)^2} = \int \frac{dx}{x}
    -\frac1{v+1} = \ln C|x|
    v = -\frac1{\ln C|x|} - 1 \Rightarrow y = -\frac{x}{\ln C|x|} - x

    So the solution is y = \frac{x}{A - \ln|x|} - x where A = - \ln C, where A is as arbitrary as C is.*


    *- These are Mr.Fantastic's words... I like it.
    Follow Math Help Forum on Facebook and Google+
« Applications of Calculus | calculus involving perimeter and diameter - any help very appreciated »

Similar Math Help Forum Discussions

  1. System of second order linear homogenous differential coupled equations
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: April 23rd 2010, 07:11 AM
  2. Second order linear homogenous Differential Equation with constant coefficients
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: March 31st 2010, 12:24 AM
  3. First Order Homogenous Differential Equation
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 10th 2010, 06:05 AM
  4. homogenous differential equation help
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: May 31st 2009, 04:13 PM
  5. linear homogenous equation, simple question.
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 25th 2009, 07:08 PM

Search Tags


/mathhelpforum @mathhelpforum