A radioactive substance decays according to :

$$x' = -ax$$

where $a>0$ is a constant. After $2$ days there are $1,000$ grams and after $7$ days there are $300$ grams. How many grams were there initially?

I'm unsure how I'd go on with doing this, any help would be greatly appreciated...

Cheers.

EDIT: The step I got upto was separation of variables/integration:

$$x = e^{-at}$$

• Hint: your equation says that $dx/dt = -ax$ so that $dx/x = -adt$. Commented Dec 11, 2013 at 22:58
• Just what I was thinking! Separation of variables followed by integration :) please let me know if that is correct so far, thank you Eric! Commented Dec 11, 2013 at 23:00
• Exactly. If you get stuck the answer is another very good hint. Commented Dec 11, 2013 at 23:02
• Okay, so doing so, I get: x = e^(-at). How would I make the 2 equations though now since I have the data for 2 days = 1000 grams and 7 days = 300 grams. Thanks again! Commented Dec 11, 2013 at 23:14
• To get multicharacter exponents, put them in braces. e^{-at} gives $e^{-at}$ This works for subscripts and items in fractions, as well. Commented Dec 11, 2013 at 23:56

Can you solve the differential equation? You should have a solution with two constants-the initial amount and $a$. The two pieces of data give you two equations in two unknowns to find these constants.
Added: your solution needs a constant of integration. The solution should be $x=c\cdot e^{-at}$. Now plug in the data you are given: $$1000=c\cdot e^{-2a}\\300=c \cdot e^{-7a}$$ Now solve those for $a,c$ and $x(0)$ is just $c$
• Yes, if you divide them, $c$ goes away and you can solve for $a$. Put that $a$ into one and you can solve for $c$. Commented Dec 12, 2013 at 2:00