Results 1 to 2 of 2

Thread: integration general solution

  1. #1
    Member
    Joined
    Apr 2006
    Posts
    91

    Question integration general solution

    Hi i have an integration problem. basically i am going through an example...the equation i want to find the gen solution is shown below..i have also shown the gen solution as well...but i cant understand how they got to that answers


    $\displaystyle \xi_{yy}$ + $\displaystyle (1/y)\xi_{y}$=0

    The general solution is given by

    $\displaystyle \xi$ = $\displaystyle A(x)\ln(y) + B(x)$

    can some one explain how they got to this general soution..thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by dopi View Post
    Hi i have an integration problem. basically i am going through an example...the equation i want to find the gen solution is shown below..i have also shown the gen solution as well...but i cant understand how they got to that answers


    $\displaystyle \xi_{yy}$ + $\displaystyle (1/y)\xi_{y}$=0

    The general solution is given by

    $\displaystyle \xi$ = $\displaystyle A(x)\ln(y) + B(x)$

    can some one explain how they got to this general soution..thanks
    The function $\displaystyle \xi$ depends on $\displaystyle x$ and $\displaystyle y$ but the equation only involves $\displaystyle y$, hence it can be seen as an ordinary differential equation.

    Let $\displaystyle x$ be fixed. Write $\displaystyle f(y)=\xi(x,y)$. Then $\displaystyle f$ is a function of 1 variable and $\displaystyle f''(y)+\frac{1}{y}f'(y)=0$ for every $\displaystyle y$. Let $\displaystyle g=f'$, so that $\displaystyle g'(y)+\frac{1}{y}g(y)=0$. This is a linear first order differential equation: we know how to solve it. Since $\displaystyle \int \frac{dy}{y}=\ln y$, there exists $\displaystyle A\in\mathbb{R}$ such that $\displaystyle g(y)=A e^{-\ln y}=\frac{A}{y}$. Hence, by integration, $\displaystyle f(y)=A\ln y + B$ for some $\displaystyle B\in\mathbb{R}$.
    However, the "constants" $\displaystyle A$ and $\displaystyle B$ depend on $\displaystyle x$ which was fixed before, hence we should write $\displaystyle A(x)$ and $\displaystyle B(x)$: $\displaystyle \xi(x,y)=f(y)=A(x)\ln y+B(x)$. This is it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. General Solution of a differential solution
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: Sep 11th 2010, 02:49 AM
  2. Finding the general solution from a given particular solution.
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: Oct 7th 2009, 01:44 AM
  3. General Solution
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 26th 2009, 07:17 AM
  4. find the general solution when 1 solution is given
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: Mar 4th 2009, 09:09 PM
  5. General solution
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: Feb 8th 2009, 07:06 AM

Search Tags


/mathhelpforum @mathhelpforum