Results 1 to 2 of 2

Math Help - integrating factor proof/ partial derivative wrt xy

  1. #1
    Junior Member
    Joined
    Oct 2009
    Posts
    39

    integrating factor proof/ partial derivative wrt xy

    The full question is:
    show that if \frac{(N_{x} - M_{y})}{(x*M-y*N)}=R where R depends only on xy, then the differential equation M+N*y'=0 has an integrating factor of the form U(xy). Find a general formula for this integrating factor.

    Using an equation from the text for U (for which finding a solution, U, means that the differential equation above is exact) I have:
    U= \frac{N*U_{x}-M*U_{y}}{M_{y}-N_{x}}
    then substituting in
    M_{y}-N_{x}=(yN-xM)*R
    and equating coefficients of N and M respectively,
    y*U*R=U_{x} and x*U*R=U_{y}

    Is there some way that I can find the derivative of U wrt xy using the partial derivatives of U wrt x and y that I have above? Then I could integrate wrt xy to find U, right?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    14
    \mu(x,y)M(x,y)dx+\mu(x,y)N(x,y)\,dx=0.

    we have \mu(x,y)=h(xy), and by the exact condition we have \displaystyle\frac{\partial \mu }{\partial y}M-\frac{\partial \mu }{\partial x}N=\left( \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} \right)\mu , now \mu_x=h'(xy)y and \mu_y=h'(xy)x, so \displaystyle\frac{1}{xM-yN}\left( \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} \right)=\frac{h'(xy)}{h(xy)}=R(xy), put z=xy and solve that little ODE and you got your integrating factor.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Integrating factor
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: February 21st 2011, 11:48 PM
  2. Partial Derivative Proof
    Posted in the Calculus Forum
    Replies: 9
    Last Post: August 19th 2010, 05:32 PM
  3. Partial Derivative of Polar Coordinates Proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 7th 2010, 09:02 AM
  4. integrating factor
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: December 23rd 2009, 08:52 AM
  5. Replies: 6
    Last Post: September 2nd 2008, 04:49 PM

Search Tags


/mathhelpforum @mathhelpforum