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Homework Help: First order linear D.E. involving i

  1. Mar 11, 2013 #1
    1. The problem statement, all variables and given/known data

    I am familiar with the standard method of obtaining a solution to a first-order, linear D.E. (i.e. using an integrating factor). However, consider the D.E. if'(x) = qf(x). It seems (after recasting the equation in the proper form) the solution suggested by the above method is f(x) = exp(iqx). However, this solution does not satisfy the original D.E. The correct solution seems to be exp(-iqx). I am curious as to why the method isn't working here. Furthermore, what general branch(s) deals with such topics if not general O.D.E.s?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 11, 2013 #2


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    Homework Helper

    If q is a (complex) constant, and i represents the square root of -1, then this is just a simple separable ODE.

    ##if'(x) = qf(x)##

    ##\frac{f'(x)}{f(x)} = \frac{q}{i} = -iq##

    Integrating both sides wrt x,

    ##\ln(f(x)) = -iqx + \ln C##

    ##f(x) = Ce^{-iqx}##

    where C is an arbitrary (complex) constant.

    So what did you do to get the answer involving ##e^{iqx}##?
  4. Mar 11, 2013 #3
    Ah, I see. I was using an integrating factor:

    That is, for equations of form

    y'(x) + h(x)y(x) = g(x)

    obtaining a solution y = y(x) via y = exp(∫h(x)dx).

    But this doesn't seem to yield the same answer.
  5. Mar 12, 2013 #4


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    Homework Helper

    Sure it does. Go through the working more carefully (you should show it here).

    What's the integrating factor here?
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