# General initial value problem (DE's)

#### prace

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

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

2. Relevant equations
**See attempt**

3. The attempt at a solution

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

so, $$A=A_0e^{kt}$$

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

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

Thanks.

#### Tomsk

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

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