Solve the non-homogeneous diusion equation ut = 2uxx + 4x ux (0, t) = 0 u (1, t) = 1 u (x, 0) = x 2 (a) Separate the problem into boundary and interior parts. (b) Find all the eigenfunctions, and transform the interior PDE. (c) Solve the resulting ODEs and invert to write the solution as a series. (d) Plot a 5-term series solution for times t = {0.0, 0.1, 0.4, 0.9, 1.6, 2.5} . For a bonus point, create and upload a 51-frame movie spanning the time domain t ∈ [0, 2.5].

Amachine that produces a special type of transistor (a component of computers) has a 2% defective rate. the production is considered a random process where each transistor is independent of the others. (a) what is the probability that the 10th transistor produced is the first with a defect? (b) what is the probability that the machine produces no defective transistors in a batch of 100? (c) on average, how many transistors would you expect to be produced before the first with a defect? what is the standard deviation? (d) another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. on average how many transistors would you expect to be produced with this machine before the first with a defect? what is the standard deviation? (e) based on your answers to parts (c) and (d), how does increasing the probability of an event a↵ect the mean and standard deviation of the wait time until success?