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MATH 215/255, Homework 7 d 1. Consider the statement: “ dt u (t − a) = δ(t − a) ”. Give an argument why this is the case. 2. Solve x00 = δ(t − 1), x(0) = 1, x0 (0) = 0. Sketch the graph of the resulting function. 3. Solve x00 + x = δ(t − π) − δ(t − 2π), x(0) = 0, x0 (0) = 1. Sketch the graph of the resulting function. 4. Find the general solution to the system x0 = Ax, where A is as specified below. Make sure to write the solution in purely real form. 1 0 (a) A = . −4 3 1 −5 (b) A = . 1 −3 1 0 0 (c) A = −2 1 2 −2 −2 1 −5 −3 (d) A = 3 1 5. A 3x3 real has three eigenvalues. One of them is λ1 = −1 and its corresponding eigenvector matrix 1 0 is v1 = 1 . Another is λ2 = 1 + i and its corresponding eigenvector is v2 = 2 . (a) What i 0 is the third eigenvalue and its corresponding eigenvector? (b) Find the general solution to x0 = Ax. 1 (c) Find the solution to x0 = Ax subject to initial condition x (0) = 0 . 0 6. Two tanks are connected by two pumps: one pump pushes the liquid from tank A to tank B at the rate of 6 liters/hour while the other pushes from tank B to tank A at the same rate. Initially, both tanks contain 2 liters of liquid and tank A contains pure water while tank B has a mixture of 80% water and 20% pollutant. (a) Find the concentration of pollutant in tanks A and B after ten minutes. (b) Find the concentration of pollutant in tanks A and B after a very long time. 1