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6 Nov 2019
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Tank A contains 10 gallons of a solution in which 5 oz of salt are dissolved. Tank B contains 20 gallons of a solution in which 6 oz of salt are dissolved. Salt water with a concentration of 2 oz/gal flows into each tank at a rate of 4 gal/min. The fully mixed solution drains from Tank A at a rate of 3 gal/min and from Tank B at a rate of 5 gal/min. Solution flows from Tank A to Tank B at a rate of 1 gal/min. Let x(t) = [x1(t) x2(t)], where x1(t) (respectively, x2(t)) is the amount of salt in Tank A (resp., Tank B) after time t. Write down a system of ODEs (including the initial condition x(0)) that models this situation, and write it in matrix form: x' = Ax + b, x(0) = c. What is the steady-state solution, xss? Write down the related homogeneous equation and solve it. Find the general solution the orginal system of differential equations modeling the tanks. Plug in t = 0 and find the particular solution. In each of the next four problems, the eigenvalues and eigenvectors of a matrix A are given. Consider the corresponding system x' = Ax. Without using a computer, draw each of the following graphs. Sketch a phase portrait of the system. Sketch the trajectory passing through the initial point (2,3). lambda1 = -4, v1 = [-1 2]; lambda2 = -1, v2 = [1 2]. lambda1 = 4, v1 = [-1 2]; lambda2 = -1, v2 = [1 2]. lambda1 = -4, v1 = [-1 2]; lambda2 = 1, v2 = [1 2]. lambda1 = 4, v1 = [1 2]; lambda2 = 1, v2 = [1 -2]. Find the general solution for each of the given systems in terms of real-valued functions, and draw a phase portrait. Describe the behavior of the solutions as t rightarrow infinity. x' = [ ]x x' = [ ]x x' = [ ]x Show transcribed image text
Please helpppppppp! 5 stars for detailed answer!
Tank A contains 10 gallons of a solution in which 5 oz of salt are dissolved. Tank B contains 20 gallons of a solution in which 6 oz of salt are dissolved. Salt water with a concentration of 2 oz/gal flows into each tank at a rate of 4 gal/min. The fully mixed solution drains from Tank A at a rate of 3 gal/min and from Tank B at a rate of 5 gal/min. Solution flows from Tank A to Tank B at a rate of 1 gal/min. Let x(t) = [x1(t) x2(t)], where x1(t) (respectively, x2(t)) is the amount of salt in Tank A (resp., Tank B) after time t. Write down a system of ODEs (including the initial condition x(0)) that models this situation, and write it in matrix form: x' = Ax + b, x(0) = c. What is the steady-state solution, xss? Write down the related homogeneous equation and solve it. Find the general solution the orginal system of differential equations modeling the tanks. Plug in t = 0 and find the particular solution. In each of the next four problems, the eigenvalues and eigenvectors of a matrix A are given. Consider the corresponding system x' = Ax. Without using a computer, draw each of the following graphs. Sketch a phase portrait of the system. Sketch the trajectory passing through the initial point (2,3). lambda1 = -4, v1 = [-1 2]; lambda2 = -1, v2 = [1 2]. lambda1 = 4, v1 = [-1 2]; lambda2 = -1, v2 = [1 2]. lambda1 = -4, v1 = [-1 2]; lambda2 = 1, v2 = [1 2]. lambda1 = 4, v1 = [1 2]; lambda2 = 1, v2 = [1 -2]. Find the general solution for each of the given systems in terms of real-valued functions, and draw a phase portrait. Describe the behavior of the solutions as t rightarrow infinity. x' = [ ]x x' = [ ]x x' = [ ]x
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