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Differential Equation using Integrating Factor

  1. Sep 14, 2009 #1
    1. The problem statement, all variables and given/known data

    The ODE is

    (1+x)*dy/dx - xy = x+x^2

    2. Relevant equations

    The method of solution is to be through the use of the integration factor.

    3. The attempt at a solution

    First, I divided each side by (1+x) to produce

    dy/dx - xy/(1+x) = x

    then factor out the x on the LHS to produce

    dy/dx - x * (y/(1+x)) = x

    then divide both sides by x to produce

    dy/dx - y/(1+x) = 0

    now move the -y/(1+x) to the RHS to produce

    dy/dx = y/(1+x)

    That is all the farther I have progressed. Am I correct so far?

    Thanks
    Matt
     
    Last edited: Sep 14, 2009
  2. jcsd
  3. Sep 14, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Yes, you are correct so far by what progress is that? How does that help you find the integrating factor?

    The left side of your equation is (1+x)(dy/dx)- xy. The whole point of an "integrating factor" is that it is a function u(x) such that u(x)(1+x)(dy/dx)- uxy= d(u(x)(1+x)y)/dx.
    Go ahead and use the product rule on the right side. That gives u(1+x)(dy/dx)+ u(1)y+ (du/dx)(1+x)y and that must be equal to u(x)(1+x)(dy/dx)- uxy. Well, the "u(1+x)(dy/dx)" cancels immediately leaving u(x)y+ (1+x)y(du/dx)= -uxy so (1+x)y(du/dx)= -xy(1+ u). Now the "y"s cancel so (1+x)(du/dx)= -x(1+u) which is a separable equation. du/(1+u)= -dx/(x(1+x)). Integrate that to find the integrating factor.
     
  4. Sep 14, 2009 #3
    By progress, I was meaning the reformatting of the original equation. I believed that this equation is seperable but the assignment was to use the integrating factor method.

    I would rather take the path of the seperable equation.

    Starting with

    dy/dx = y/(1+x)

    multiplying both sides by dx produces

    dy = y/(1+x)dx

    dividing both sides by y produces

    dy/y = dx/(1+x)

    integrating both sides produces

    ln(y) = ln(1+x) + c

    Is this correct so far?


    Now proceeding forward with your instructions to integrate

    du/(1+u) = -dx/(x(1+x)

    upon integration of both sides

    ln(1+u) = -ln(x)+ln(1+x)

    now the integrating factor would be

    e^(ln(1+u))

    upon integration would be

    1+u

    Am I correct so far?

    Thanks
    Matt
     
    Last edited: Sep 14, 2009
  5. Sep 14, 2009 #4
    OK. I think I have the solution.

    With the integrating factor of 1+x and using the equation

    dy/dx -y/(1+x) = 0

    multiplying both sides by the integrating factor produces

    (1+x)*dy/dx - (1+x)*y/(1+x) = 0

    which is equal to

    (1+x) * dy/dx - y = 0

    upon integrating

    (1+x)*y = c

    solving for y produces

    y = c/(1+x)

    Am I correct?

    Thanks
    Matt
     
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