In the following, we will use the following notations: 2xe, 21, 2te for the decay constants, Nxe,...

In the following, we will use the following notations: 2xe, 21, 2te for the decay constants, Nxe, Ni, NTe for the isotope population; Xe, F?, F?e for each isotope fission yields (constant production through fission process in ion/s) and Lxe, Li, Lte the Laplace transform of Nxe, NI, NTe. Xe134 Xe136 2.36E21 0+ B- Xe137 3.818 m 7/2- 0+ Xe135 9.14 h 3/2+ * B 1134 52.5 m 10A 1133 20.8 h 7/2+ 8.9 1135 6.57 h 7/2+ B 1136 834s 1. Compare the decay constant (a) and the so-called "half-life” which is defined as the value of t when e-ht=1/2. Give the 2xe, 21, 2 Te as a function of the respective half-lives detailed on the previous figure. ?? Tel32 3.204 d 0+ Te133 12.5 m (3/2+) Te134 41.8 m 0+ Te135 19.0 s (7/2-) P B- IP 2. Explain why the differential equation that represents the time evolution of 135 Te population can be written: dNte (t) = -AreNre(t) + Ore dt 3. Using a similar argument, write 1351 and 135Te population differential equations to obtain a system of coupled first order linear equations. This system is known as a Bateman equation system. There are several methods available to solve this well known system of equations. One of them uses the Laplace transform. 4. In a few words, explain the main general interest of using integral transforms to solve differential equations problems. 5. Using the Laplace transform definition, rewrite the system such as: sLle SLI SLxe -XTelte + ??? -AILI + Telte = -xelxe + ALI + + 1 8 Xe 6. Using inverse Laplace transform, find the population of Te, I and Xe. 7. Comment on the simplicity of the method has compared to a more standard way of solving differential equations.