Two tanks are interconnected. Both tanks initially contain 90 L of salt solution. The first tank initially contains 3 kg of dissolved salt and the second tank initially contains 2 kg of dissolved salt. Pure water enters the first tank at a rate of 4.5 Liters per minute. Salt solution from the first tank flows into the second tank at a rate of 6 Liters per minute and salt solution from the second tank is pumped back into the first tank at a rate of 1.5 Liters per minute. Salt solution leaves the system through the second tank at a rate of 4.5 Liters per minute. a) Represent the situation described above as a linear system of differential equations with initial conditions. Use Q, for the mass of salt in the first tank after t minutes and O, for the mass of salt in the second tank after t minutes.

Two tanks are interconnected. Both tanks initially contain 90 L of salt solution. The first tank initially contains 3 kg of dissolved salt and the second tank initially contains 2 kg of dissolved salt. Pure water enters the first tank at a rate of 4.5 Liters per minute. Salt solution from the first tank flows into the second tank at a rate of 6 Liters per minute and salt solution from the second tank is pumped back into the first tank at a rate of 1.5 Liters per minute. Salt solution leaves the system through the second tank at a rate of 4.5 Liters per minute. a) Represent the situation described above as a linear system of differential equations with initial conditions. Use Q, for the mass of salt in the first tank after t minutes and O, for the mass of salt in the second tank after t minutes.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

f) Use your eigenvalues and eigenvectors to write the general solution, in matrix form, to your linear
system from part (a). (Your answer should have two arbitrary constants).

g) Apply your initial conditions from part (a) to solve for the arbitrary constants.

Please answer a-h with clear steps

thanks
h) Take your solution out of matrix form and rewrite it in the form Q1= f(t), Q2 = g(t)

**Problem Statement:**

Two tanks are interconnected. Both tanks initially contain 90 L of salt solution. The first tank initially contains 3 kg of dissolved salt and the second tank initially contains 2 kg of dissolved salt. Pure water enters the first tank at a rate of 4.5 Liters per minute. Salt solution from the first tank flows into the second tank at a rate of 6 Liters per minute and salt solution from the second tank is pumped back into the first tank at a rate of 1.5 Liters per minute. Salt solution leaves the system through the second tank at a rate of 4.5 Liters per minute.

a) Represent the situation described above as a linear system of differential equations with initial conditions. Use \(Q_1\) for the mass of salt in the first tank after \(t\) minutes and \(Q_2\) for the mass of salt in the second tank after \(t\) minutes.

b) Rewrite your initial value problem from part (a) in **matrix form**.

c) Find the characteristic polynomial of the coefficient matrix from your matrix equation in (b).

d) Find the eigenvalues of the coefficient matrix by solving for when the characteristic polynomial is equal to 0.

e) Find the corresponding eigenvectors of the coefficient matrix.

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