Tank A contains 10 gallons of a solution in which 5 oz of saltare dissolved. Tank B
contains 20 gallons of a solution in which 6 oz of salt aredissolved. Salt water with a
concentration of 2 oz/gal flowsinto 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 TankB at a rate of 5 gal/min.
Solution flows from Tank A to TankB 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.
(a) Write down a system of ODEs (including the initial conditionx(0)) that models this
situation, and write it in matrix form: x' = Ax + b, x(0) = c.
(b) What is the steady-state solution, x_ss?
(c) Write down the related homogeneous equation and solveit.
(d) Find the general solution the orginal system of dierentialequations modeling the
tanks.
(e) Plug in t = 0 and nd the particular solution.
Tank A contains 10 gallons of a solution in which 5 oz of saltare dissolved. Tank B
contains 20 gallons of a solution in which 6 oz of salt aredissolved. Salt water with a
concentration of 2 oz/gal flowsinto 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 TankB at a rate of 5 gal/min.
Solution flows from Tank A to TankB 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.
(a) Write down a system of ODEs (including the initial conditionx(0)) that models this
situation, and write it in matrix form: x' = Ax + b, x(0) = c.
(b) What is the steady-state solution, x_ss?
(c) Write down the related homogeneous equation and solveit.
(d) Find the general solution the orginal system of dierentialequations modeling the
tanks.
(e) Plug in t = 0 and nd the particular solution.
