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Second order linear system and power series: Differential Equations

  1. Dec 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Find a third degree polynomial approximation for the general solution to the differential equation:

    [tex]\frac{d^{2}y}{dt^{2}}[/tex] +3[tex]\frac{dy}{dt}[/tex]+2y= ln(t+1)

    2. Relevant equations
    Power series expansion for ln(t+1)


    3. The attempt at a solution

    The system to the corresponding homogeneous equation [tex]\frac{d^{2}y}{dt^{2}}[/tex] +3[tex]\frac{dy}{dt}[/tex]+2y = 0

    is y(t) = k1e-t +k2e-2t

    Then I guessed[tex]\frac{ at^{3}}{3}[/tex]-[tex]\frac{bt^{2}}{2}[/tex]+ct as a solution for the original equation. Plugging this in I got a=1/2, b=2,c=2/3

    But then I still have the t[tex]^{4}[/tex], t[tex]^{5}[/tex] terms, etc left in the equation. Im not quite sure how a third degree polynomial can be a solution to this equation.
     
  2. jcsd
  3. Jan 3, 2010 #2
    Because it will be an approximation not really the solution itself.
     
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