# Problem 1 Let A be a 2 × 2 matrix such that e At = What...

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The most exciting phrase to hear in science, the one that heralds new discoveries, is not “Eureka!” but “That’s funny ...”. — Isaac Asimov (1920–1992) Problem 1. Let A be a 2 × 2 matrix such that e At = . What are the values of c and r ? parenleftBigg (1 - t ) e 2 t te 2 t parenrightBigg
Problem 2. Consider x = A x where A = parenleftbigg 2 1 6 3 parenrightbigg . (a) Find the general solution. (b) Find e At . (c) Find a particular solution to x = A x + .
2/ t 2 Problem 3. The mixtures in three tanks T 1 ,T 2 ,T 3 are kept uniform by stirring. Brine containing 2 lb of salt per gallon enters the first tank at 15 gal/min. Mixed solution from T 1 is pumped into T 2 at 10 gal/min and from T 2 into T 3 at 10 gal/min. Each tank initially contains 10 gal of pure water. Denote by x i ( t ) the amount (in pounds) of salt in tank T i at time t (in minutes). Derive a system of linear differential equations for the x i .