What is the correct way to solve this ordinary differential equation decay problem?

Solution 1:

Plug in $t=0$ into the solution you get, you will have $x=Ae^0=A$. So you have value for $A$. For $k$, as you have shown, you have $0.9x=xe^{-200k}$. So you have $$-200k=\ln(0.9).$$ When you consider the case after 1000 years, you have $$-1000k=5\ln(0.9)=\ln(0.9^5).$$ So you will see $$e^{-1000k}=0.9^5.$$ This is the percent you are looking for.

Solution 2:

$A$ is the starting amount of radium and $x$ is the current amount of radium. Therefore, $0.9$ is $\frac{x}{A}$. Use that to solve for $k$, and use the same idea to then extrapolate to a different $\frac{x}{A}$ for a different $t$.