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Homework Help: 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?

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


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    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


    upon integration would be


    Am I correct so far?

    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?

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