A given mass x of a radioactive element obeys the following differential equation in time:
x˙ =λx, where λ is a constant describing the rate of decay. (a) Write down the solution x(t) to the differential equation. (b) Plot the solution for an initial condition x(0)=2 from time t=0 to t=5 for λ= −5,−1,0,0.01, and 0.1 . Please plot these all on the same figure. Label your axes and include a legend. (c) The half-life T is defined as the time it takes for the material to be reduced to half of its mass through radioactive decay. The half-life of uranium- 238 is 4.468 billion years. What is the corresponding value of λ ? (d) If you start with 50 kg of uranium-238, how long until you only have 10 kg left?