1. For our model of nuclear decay (N >> 1) we have N(t) = N_oe^-t/τ, where τ is the e-folding time of our decay or the mean lifetime. Derive an expression for the half-life (λ) of an isotope in terms of τ.
   Since N = N_oe^-t/τ, N(λ) = N_o/2. Solving for λ, we find λ = τln 2, conversely, τ = λ/ln 2.

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For the following problems, use the Euler method ode solver you developed in Lab #1. Be sure to include the following with your homework solution:

• brief discussion (your approach, assumptions, observations);
• complete code listings (with comments);
• graphical results (comparison of analytic results with your numerical simulation);
• a discussion of errors.
2. Select a radioactive isotope (other than ²³⁵U or ²³¹Th). Find it's theoretical τ and λ. Simulate it's decay beginning with a suitable N_o. Verify that the half-life is indeed λ.
   A code to run & solve this problem is given by hw2_2.m. It makes use of the generic ODE integrator I mentioned in lecture – check it out to see how it can work for your problems. Since there are too many isotopes to deal with, I've normalized things by letting n = N/N_o and let tt = t/τ – now the number is expressed as a fraction of the original population and time is given in half-lives. We expect n(1) = 0.5.
   
   
   In the left panel we see that the results are close to, but not exactly, the prediction. In the right panel we see that the relative error is a function of Δt and varies linearly with it. To realize a 1% error in half-life prediction, we need take Δt to be about 4% of λ.



3. G-N §§1.1
   For dv/dt = -g, and Euler step is given by v(t+Δt) ≈ v(t) + dv/dt|_tΔt = v(t) - gΔt. The exact solution follows from
   dv = -g dt
   v - v_o = -g(t-t_o)
   v(t+t_o) = v(t) - gΔt — an exact match to the Euler method. Note that this exact match is not a function of Δt.



4. G-N §§1.3 (For this problem, consider the errors in terms of the terminal velocity, v_t = a/b.)
   See the code hw2_4.m for an example of how to approach this problem (you'll also need the terminalVelocity.m differential equation code). Consider the plots below.
   
   
   Even w/out knowing the exact solution (see display case for a derivation), we see that the approximate Euler solutions begin to converge on a common curve. In the right panel, I zoom in on t ∈ [5,7] and look at a log plot of the relative errors in finding v_t = a/b. Again, with smaller Δt, we start to converge quickly on the actual value. Effectively, we're "at" v_t after 7 s, to w/in 1 part in 1000.

Extra Credit. Consider the case of exponential growth, where τ < 0. This is just the more general case of exponential change where τ ∈ ℜ, for which our original solution and numerical approach are valid. Compare the performance of the Euler method for long times using a simulation of ±τ. Pay particular attention to the errors.
This situation is discussed in Lab #2, §§4.