Results 1 to 5 of 5

Math Help - [SOLVED] differential equation given integrating factor

  1. #1
    Member
    Joined
    Nov 2009
    Posts
    151

    [SOLVED] differential equation given integrating factor

    Show that given function μ is an integrating factor and solve the differential equation..

    y^2 dx + (1 + xy) dy = 0 ; μ(x) = e^xy

    let M = y^2
    N = (1 + xy)

    dM/dy = 2y dN/dx = y hence, not exact equation.

    times μ(x) = e^xy to the not exact equations...

    2y(e^xy) dx + y(e^xy) dy = 0

    let M = 2y(e^xy)
    N = y(e^xy)

    dM/dy = 2(e^xy) + 2y(e^y) ---> apply product rule

    dN/dx = 0(e^xy) + y(e^y) ---> apply product rule

    the problem is.. the equations still not the exact equations..
    How to proceed?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    10,969
    Thanks
    1011
    Quote Originally Posted by nameck View Post
    Show that given function μ is an integrating factor and solve the differential equation..

    y^2 dx + (1 + xy) dy = 0 ; μ(x) = e^xy

    let M = y^2
    N = (1 + xy)

    dM/dy = 2y dN/dx = y hence, not exact equation.

    times μ(x) = e^xy to the not exact equations...

    2y(e^xy) dx + y(e^xy) dy = 0

    let M = 2y(e^xy)
    N = y(e^xy)

    dM/dy = 2(e^xy) + 2y(e^y) ---> apply product rule

    dN/dx = 0(e^xy) + y(e^y) ---> apply product rule

    the problem is.. the equations still not the exact equations..
    How to proceed?
    y^2\,dx + (1 + xy)\,dy = 0

    y^2\,dx = -(1 + xy)\,dy

    y^2\,\frac{dx}{dy} = -(1 + xy)

    y^2\,\frac{dx}{dy} = -1 - xy

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

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


    Now this is a linear DE in x.

    So the integrating factor is e^{\int{y^{-1}\,dy}} = e^{\ln{y}} = y

    Multiplying through by the integrating factor gives:

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

    \frac{d}{dy}(xy) = -y^{-1}

    xy = \int{-y^{-1}\,dy}

    xy = -\ln{|y|} + C

    x = -y^{-1}\ln{|y|} + Cy^{-1}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2009
    Posts
    151
    got it.. so... the integrating factor given, is not the correct integrating factor... right?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    10,969
    Thanks
    1011
    Quote Originally Posted by nameck View Post
    got it.. so... the integrating factor given, is not the correct integrating factor... right?
    Unless there's a way to make it a linear equation in y. However, I couldn't find one, and since you can't rewrite the solution as y in terms of x, I doubt that you can...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Nov 2009
    Posts
    151
    thanks Prove It!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solving Differential Equations with An Integrating Factor
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: May 14th 2010, 01:09 PM
  2. [SOLVED] find integrating factor and solve the equation 3
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: March 27th 2010, 11:17 PM
  3. [SOLVED] find integrating factor and solve the equation 2
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: March 27th 2010, 07:58 PM
  4. [SOLVED] find integrating factor and solve the equation
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: March 27th 2010, 07:24 PM
  5. Differential equation - integrating factor
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: June 19th 2009, 06:18 AM

Search Tags


/mathhelpforum @mathhelpforum