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Solving a linear ODE

  1. Aug 31, 2011 #1
    1. The problem statement, all variables and given/known data
    Solve [itex]\frac{dy}{dx}[/itex] - 2y = x[itex]^{2}[/itex]e[itex]^{2x}[/itex]

    3. The attempt at a solution

    Integrating factor = e[itex]^{2x}[/itex]

    So we multiply through the given equation by the integrating factor and get:

    e[itex]^{2x}[/itex][itex]\frac{dy}{dx}[/itex] - 2e[itex]^{2x}[/itex]y = x[itex]^{2}[/itex]e[itex]^{4x}[/itex]

    Contract the left-hand side via the chain rule to get:

    [itex]\frac{d}{dx}[/itex](e[itex]^{2x}[/itex]y) = x[itex]^{2}[/itex]e[itex]^{4x}[/itex]

    Integrate both sides

    e[itex]^{2x}[/itex]y = [itex]\frac{1}{32}e^{4x}[/itex](8x[itex]^{2}[/itex]-4x+1)+C

    Now divide through by e[itex]^{2x}[/itex] and the equation definitely does not equal what Wolfram Alpha gives as the solution:

    y = [itex]\frac{1}{3}[/itex]e[itex]^{2x}[/itex]x[itex]^{3}[/itex]+Ce[itex]^{2x}[/itex]

    I checked some of the parts individually with Wolfram, such as the integration of the right-hand side and that was correct, so I'm not too sure what's causing the difference in answers.
  2. jcsd
  3. Aug 31, 2011 #2
    Grr, nvm. I forgot the minus sign in the integrating factor -.-
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