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2nd order differential equation (nonhomogenous)

  1. Jan 13, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the general solution f(t) of the differential equation: f''(t) − f'(t) − 12f(t) = 36π . How many free parameters enter the solution, and why?

    2. Relevant equations
    f = fH + fP where fH is the homogeneous solution and fP is the particular solution.

    3. The attempt at a solution
    So I've solved this equation to get: f(t) = C1e4t + C2e-3t which I have checked is right.

    My question is how may free parameters enter the solution? I'm not sure what this means - is it asking how many constants there are and the answer would therefore be 2? But I don't know why.

    Please help! Thank you.
  2. jcsd
  3. Jan 13, 2016 #2


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    First, why in the world is a question about "differential equations" posted in the "PreCalculus" section?

    Yes, the "free parameters" refer to the number of arbitrary constants. Further the basic theory here is that "the set of all solutions to an nth order linear homogeneous differential equation form a vector space of dimension 2". This is a second order differential equation so the dimension is 2- any solution can be written as a linear combination of two independent solutions giving the two constants.
  4. Jan 13, 2016 #3


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    It may be a question of terminology, but I have seen the constants be called the free parameters.
    So you have two.

    You have not solved the differential equation yet, you have found fH.
    You still need to find one fP (shouldn't be too difficult)
  5. Jan 13, 2016 #4
    Oh I forgot to add in the fP = -3π. So why are there 2 free parameters, do you know? Is it because the equation is twice differentiated leaving two constants? Thank you
  6. Jan 13, 2016 #5


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    Yes, the solution space of your second order linear homogeneous differential equation has dimension 2.
  7. Jan 13, 2016 #6

    Ray Vickson

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    Because the DE involves second-order derivatives you need two additional conditions in order to fully express the solution (besides the DE itself). For example, you need to know f(0) and f'(0) [initial-value problem] or f(t1) and f(t2) [a two-point boundary value problem], or some initial value such as f(t1) and large-t-behavior (at t →∞), etc. If I tell you the value of f(0), for example, you will still have one undetermined parameter in the solution, and that parameter may be used to match additional problem information.
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