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General initial value problem (DE's)

  1. Dec 11, 2006 #1
    1. The problem statement, all variables and given/known data
    a) Consider the initial value problem [tex]\frac{dA}{dt} = kA, A(0) = A_0[/tex] as the model for the decay of a radioactive substance. Show that in general the half-life T of the substance is [tex]T = -\frac{ln2}{k}[/tex]

    b) Show that the solution of the initial-value problem in part a) can be written as [tex]A(t) = A_02^{\frac{-t}{T}}[/tex]

    2. Relevant equations
    **See attempt**

    3. The attempt at a solution

    So I started with the given information: [tex]\frac{dA}{dt} = kA, A(0) = A_0[/tex] and turned it into a DE then solving by the separation of variables technique.

    so, [tex]A=A_0e^{kt}[/tex]

    From here I tried to divide the whole equation by [tex]\frac{A}{2}=A_0e^{kt}[/tex], but that did not seem to do anything.

    Can anyone please give me a pointer to get me started in the right direction?

  2. jcsd
  3. Dec 11, 2006 #2
    You are trying to find the time when A is half its initial value, ie [itex]A(T)=A_{0}/2[/itex]. You got the right formula for A, so you should say [itex]A(T)=A_{0}/2=A_0e^{kT}[/itex] and then solve for T.
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