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(15) w8.2 Consider a general input-output model with two compartments as indicated below. The compartments contain volumes V₁, V₂ and solution amounts x₁(t), x2(t) respectively. The flow rates (volume per time) are indicated by ri, i = 1..6. The two input concentrations (solute amount per volume) are C1, C5. 5₁. c. V3 b-d V₁ x, (+) %₂2lt) (a) What equalities between the flow rates guarantee that the volumes V₁, V₂ remain constant? (b) Assuming the equalities in a hold, what first order system of differential equations governs the rates of change for x₁(t), x2(t)? (c) Suppose r2 = r4 = r6 = 100, r3 = r5 = 200, r₁ = 0. hour C₁ = 0, c5 = 0.3; V₁ V₂ = 100 gal. Verify that the constant volumes are consistent with the rate balancing required in a. Then show that the general system in b reduces to the following system of DEs for the given parameter values: [268] 28-7³ +9 -3 (d) Solve the initial value problem in c, assuming there is initially no solute in either tank. Hint: Find the homogeneous solution; then find a particular solution which is a constant vector; and then use = p + H to solve the IVP. (e) Re-solve the IVP in part d with the direct approach based on diagonalization of the matrix 2 -3 A = in this system of DEs: let P be a matrix of eigenvector columns that diagonalizes A, i.e. AP= PD where D is the corresponding diagonal matrix with eigenvalues along the diagonal. Make the change of functions Pū(t) = (t)