[1 1. Find the general solution of X' = 0 0 3 1 x. -1 1 2. Use the following figure to construct a model for the number of pounds of salt 1(t), x2(t), and x3(t) at time t in tanks A, B, and C, respectively. Write the model in matrix form and then solve it using eigenvalue/eigenvector techniques assuming that x1(0) 15, x2(0) = 10, and 3(0) = 5. Will all of the tanks eventually be free of salt? Use your solution to justify your answer. pure water 4 gal/min A B 150 gal 4_4 200 gal mixture 4 gal/min mixture 4 gal/min C 100 gal mixture 4 gal/min 3. Consider the system tx' = 2 x, t0 with initial condition (2) Assuming solutions of the form x = tv where A, are an eigenvalue/eigenvector pair of the given matrix, use techniques similar to those used to construct solutions to the constant coefficient linear homogeneous systems to solve the given initial value problem. Write your answer as a single vector. 4. Solve the initial value problem = =3x-2 =-3y-2 Myrigh = 2y-2 with x(0) -5, y(0) = 13, z(0) = -26 using eigenvalue/eigenvector techniques. Bou 5. Consider the system x'= following values of c: (a) c (0,∞) (c) c€ (-1,0) (d) c = -1 (e) c (-0,-1) where c is a parameter. Classify the geometry and stability properties of the system for the Answers To Selected Problems: These are final answers for selected problems so you can check your work and determine if you are on the right track. To receive credit on each problem you must show all steps, fully answer the problem, and justify your answer using correct mathematical notation. x1(t) = 15e-t/50 2. x2(t)=45e-t/50-35e-2t/75 x3(t)=60e-t/50-70e-2t/75 + 15e-t/25 [10t-1-412] 212 3. x(t)=20t-1 x(t) e 2 (sint - 5 cost) 4. y(t) 13e 2 (sint + cost) 2(t)=-26e-2 cost