(a) The radioactive decay of polonium 218 to bismuth 214 is given by 218Po → 214Pb → 214Bi, where the first reaction rate is k1 = 0.5 s−1, and the second reaction rate is k2 = 0.06 s−1.

(i) Write down the differential equations representing this system.

Solve the ODEs.

(ii) Determine the amount of each substance after 20 seconds given that the initial amount of 218Po was one unit. Assume that the initial amounts of the other two substances were zero.
(iii) Plot solution curves against time for each substance.
(iv) Plot a trajectory in three-dimensional space.
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(b) Plot the limit cycle of the system dx dt = y+0.5x(1−0.5−x2−y2), dy dt = −x+0.5y(1−x2−y2).
Find the approximate period of this limit cycle.
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