# Differential Equation - Radioactive Decay

• Nov 4th 2008, 02:06 AM
jeta
I have a pretty hard differential equation here:

In the nuclear medicine department of a hospital, a radioactive isotope is being produced. The isotope is produced at the rate of 5 milligrams per minute, but has a half-life of only 20 minutes. Write differential equations to find the amount of the isotope present after an hour of production.

Now this one's a bit wierd...

I'm fine with an 'initial amount' problem but when there is a constant production, well it's a bit hard.

What I think to put is

dM/dt = 5-how much is lost (0.5M/20?????)

• Nov 4th 2008, 02:42 AM
mr fantastic
Quote:

Originally Posted by jeta
I have a pretty hard differential equation here:

In the nuclear medicine department of a hospital, a radioactive isotope is being produced. The isotope is produced at the rate of 5 milligrams per minute, but has a half-life of only 20 minutes. Write differential equations to find the amount of the isotope present after an hour of production.

Now this one's a bit wierd...

I'm fine with an 'initial amount' problem but when there is a constant production, well it's a bit hard.

What I think to put is

dM/dt = 5-how much is lost (0.5M/20?????)

$\displaystyle \frac{dm}{dt} = 5 - \lambda m$ subject to the boundary condition m = 0 when t = 0 and where $\displaystyle 20 = \frac{\ln 2}{\lambda}$.