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Homework Help: How to integrate (e^t)*tan^2(t)

  1. Oct 21, 2012 #1
    1. The problem statement, all variables and given/known data

    I am in the process of solving second order differential equation by the variation of parameters method. I need to calculate ∫(et)*(tant)2 dt

    2. Relevant equations
    (tanx)2 = (secx)2-1
    ∫(secx)dx= lnlsecx*tanxl

    3. The attempt at a solution
    I have used the trig identity to get this: ∫(et*((sect)2-1)dt
    = ∫et*(sect)2 dt -∫(et) dt
    =∫et*(sect)2 dt -et
    but am stuck here. Any help would be greatly appreciated. Thanks!
    Last edited: Oct 21, 2012
  2. jcsd
  3. Oct 21, 2012 #2


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    I think it's not expressible in terms of elementary functions. Find another method for your ODE.
  4. Oct 21, 2012 #3
  5. Oct 21, 2012 #4
    The problem explicitly asks to solve using variation of parameters. Perhaps I made a mistake in the algebra somewhere before this step; I will go to my professor. Thanks for your input, now I won't waste a hour or two staring at and trying to solve this integral.
  6. Oct 21, 2012 #5


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    State the ODE, please. There might be a mistake somewhere, or your professor really likes Gauss hypergeometric functions.
  7. Oct 21, 2012 #6
    I doubt it; I probably made a mistake. Here is the problem:

    Find the general solution by variation of parameters:
    y'' + y = (tanx)2

    For these, I know you find the solution by finding and adding together the solution to the homogeneous equation with that of your particular solution.

    I found the solution to the homogeneous equation to be:
    yh = C1*e-t+C2

    Then I needed to find the particular solution which I knew would be in the form u1*y1 + u2*y2
    u1 = -∫ [itex]\frac{y2*g}{W[y1,y2]}[/itex] dt
    u2= ∫ [itex]\frac{y1*g}{W[y1,y2 ]}[/itex] dt

    (g is (tanx)2, and W is the Wronskian)
  8. Oct 21, 2012 #7


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    It's not right. y''+y=0 doesn't have the solution you wrote as y_h. No way.
  9. Oct 21, 2012 #8
    Ah! I see. I was using reduction of power method and messed up. I had gotten r2 + r =0 whereas it should be r2+1=0 . Therefore I should get an imaginary root, and the solution is yh= C1*sint + C2*cost
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