ODE with integrating factor NEED HELP

  1. 1. The problem statement, all variables and given/known data

    [tex] (2x+y^2) dx +4xy dy=0,y(1)=1 [/tex]

    2. Relevant equations

    3. The attempt at a solution
    I'm having trouble finding the correct integrating factor, been playing with it for an hour and have made NO progress so need help.

    [tex] \delta P/\delta y=2y [/tex]
    [tex] \delta Q/\delta x=4y [/tex]

    so its not exact and I need to find an integrating factor.

    [tex] R=1/Q( \delta P/\delta y-\delta Q/\delta)=-1/2x [/tex]
    [tex] F(x)= exp \int R(x) dx [/tex]
    [tex] \int -1/(2x) = -ln(abs(x))/2 [/tex]

    This integrating factor isn't going to make my ode exact though and I can't figure out what I have stuffed up???

    Thanks for any help.
  2. jcsd
  3. LCKurtz

    LCKurtz 8,183
    Homework Helper
    Gold Member

    Don't quit yet, you almost have it. Take x > 0 to dispense with the || signs:

    [tex]R(x) = e^{\int-\frac 1{2x}dx}= e^{-\frac 1 2 \int\frac 1 x}\, dx
    = e^{-\frac 1 2 \ln x} = e^{\ln(x^{-\frac 1 2})} = x^{-\frac 1 2}[/tex]

    Also, a TeX note, you can use the partial symbol

    [tex]\partial Q/\partial x[/tex]

    but subscripts are even easier: Qx or [itex]Q_x[/itex].
  4. Thanks LCKurtz I now have a general and particular solution, would you mind checking my results:
    [tex] u(x,y)=4x^{3/2}+2*\sqrt(x)*y^2+y=c [/tex]
    [tex] c=13/3 [/tex]
    [tex] 13/3=4x^{3/2}+2*\sqrt(x)*y^2+y [/tex]


    ps should be 4x to the power of 3/2 , doesn't seem to like my tex.
  5. LCKurtz

    LCKurtz 8,183
    Homework Helper
    Gold Member

    You should be able to check that yourself. Check that du = Pdx + Qdy with the exact form and that (x,y)=(1,1) works. If not, try to fix it.
  6. I don't know what you mean by Check that du = Pdx + Qdy, du as in the differential of
    [tex] 4x^{3/2}+2*\sqrt(x)*y^2+y [/tex]

    also should [tex]
    [/tex] = 13/3? because it =s 7??

    am I doing something wrong with my checking or is my solution wrong?

  7. The whole point of finding an integrating factor is converting your differential equation into an exact one. What LCKurtz is suggesting is that you use the partial derivative test to see if it is indeed exact.
  8. LCKurtz

    LCKurtz 8,183
    Homework Helper
    Gold Member

    After you multiplied by the integrating factor x-1/2 you should have gotten this DE, which you didn't show:

    [tex](*)\ (2x^{\frac 1 2}+ 2x^{-\frac 1 2}y^2)dx+4x^{\frac 1 2}y dy = 0[/tex]

    Presumably you have checked that this equation is exact and you have a proposed solution u(x,y) written above. The way to check whether your solution is correct is to calculate du = uxdx + uydy and see if it agrees with the exact equation (*) above.

    Once you have that you need to check that x = 1, y = 1 works in your final answer.
  9. Decided to practice a bit on this one. Here's what I've got:

    1. Orig. DE is exact with the integ. factor: [tex]\mu=1/\sqrt{x}[/tex] because [tex]\frac{\partial(\frac{2x+y^2}{\sqrt{x}})}{\partial y}=\frac{2y}{\sqrt{x}}=\frac{\partial(4\sqrt{x}y)}{\partial x}\Longleftrightarrow\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}[/tex]

    2. To solve for u, we have: [tex]u(x,y)=\int\frac{2x+y^2}{\sqrt{x}}dx+g(y)=\frac{4x^{3/2}}{3}+2\sqrt{x}y^2+g(y)[/tex]

    3. To find g(y), we take the partial derivative of the newly found function u(x,y) in (2) with respect to y and set it equal to Q, that is: [tex]\frac{\partial u}{\partial y}=4\sqrt{x}y+\acute{g(y)}\Longrightarrow 4\sqrt{x}y+\acute{g(y)}=4\sqrt{x}y \Longrightarrow \acute{g(y)}=0[/tex]

    4. So, after integrating the result in (3), we find that: [tex]g(y)=c_{1}[/tex] Thus, the overal function u(x,y) takes the form: [tex]u(x,y)=\frac{4x^{3/2}}{3}+2\sqrt{x}y^2+c_{1}=c_{2}[/tex] or, combining the constants, we have: [tex]\frac{4x^{3/2}}{3}+2\sqrt{x}y^2=c[/tex]

    5. If y(1)=1, the constant c works out to be c=10/3, so that the unique solution becomes: [tex]\frac{4x^{3/2}}{3}+2\sqrt{x}y^2=\frac{10}{3}[/tex]

    6. A simple check shows that: [tex]du=u_{x}dx+u_{y}dy=(\frac{4}{3}\cdot\frac{3}{2}x^{1/2}+2\frac{1}{2}x^{-1/2}y^2)dx+(2\cdot2\sqrt{x}y)dy=(\frac{2x+y^2}{\sqrt{x}})dx+(4\sqrt{x}y)dy[/tex] which is the original DE multiplied by the integrating factor.

    7. I learned some simple TEX commands o:)
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