Results 1 to 3 of 3

Math Help - Simple differential equation problem

  1. #1
    Senior Member
    Joined
    Jan 2009
    Posts
    290

    Simple differential equation problem

    (1+x^2)y\frac {dy} {dx}+(1+y^2)x=0

    This is the steps the solution book gave.

    Rearranging the original equation

    \frac {y} {1+y^2} dy=-\frac {x} {1+x^2} dx

    Integrating both sides

    \frac {1} {2} \ln(1+y^2)=-\frac {1} {2} \ln(1+x^2)+c

    Usually in this step I solve for y (which is what the book has been doing before too), and I get rid of the ln function. But in this question my solution book gives this.

    \ln[(1+x^2)(1+y^2)]=2c

     (1+x^2)(1+y^2)=e^{2c}

    Why?

    The final answer is

    (1+x^2)(1+y^2)=k after replacing k with e^{2c}

    Is this the same thing as solving for y, but written in a different form?
    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 chengbin View Post
    (1+x^2)y\frac {dy} {dx}+(1+y^2)x=0

    This is the steps the solution book gave.

    Rearranging the original equation

    \frac {y} {1+y^2} dy=-\frac {x} {1+x^2} dx

    Integrating both sides

    \frac {1} {2} \ln(1+y^2)=-\frac {1} {2} \ln(1+x^2)+c

    Usually in this step I solve for y (which is what the book has been doing before too), and I get rid of the ln function. But in this question my solution book gives this.

    \ln[(1+x^2)(1+y^2)]=2c

     (1+x^2)(1+y^2)=e^{2c}

    Why?

    The final answer is

    (1+x^2)(1+y^2)=k after replacing k with e^{2c}

    Is this the same thing as solving for y, but written in a different form?
    The book has decided to define the solution implicitly. If you want to make y the subject, go right ahead. It's not difficult but obviously the solution won't look as elegant.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Jan 2009
    Posts
    290
    Thanks. Just what I thought.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Simple Differential Equation
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: March 21st 2011, 06:40 PM
  2. probably very simple differential equation :D
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: September 12th 2010, 10:09 PM
  3. Replies: 2
    Last Post: March 15th 2010, 03:33 AM
  4. Simple differential equation problem
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: April 13th 2009, 07:05 AM
  5. simple differential equation
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 18th 2007, 12:13 PM

Search Tags


/mathhelpforum @mathhelpforum