General initial value problem (DE's)

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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?

Thanks.
 
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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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