In an ice tray, the water level in any particular ice cube cell will change at a rate proportional to the difference between the cell's water level and the level in the adjacent cells. (a) Argue that a reasonable differential equation model for the water levels x, y, and z in the simplified three-cell tray depicted in the accompanying figure is given by x'=y-x, y'=x+z-2y, z' =y-z. (b) Use eigenvectors to solve the system for the initial conditions x(0) = 6, y(0) = 0, z(0) = 3. (a) The first equation is reasonable because the change in the water level of the first cell, x', is proportional to the difference between the first cell's water level and the level in the adjacent cell, y-x. The constant of proportionality between x' and y-x is k = 1. The third equation is reasonable because the change in the water level of the third cell, z', is proportional to the difference between the third cell's water level and the level in the adjacent cell, y-z. The constant of proportionality between z' and y-x is k = 1. The total rate of change in the water level f the second cell, y', is dependent on the water level in the first cell and the level in the third cell. The difference between the level in the second cell and the level in the first cell is x-y. The difference between the water level in the second cell and the level in the third cell is z-y. Therefore, if y' is the total rate of change in the water level of the second cell, it is reasonable that y' is the (b) Solve the system for the given initial conditions. x(t) y(t) = z(t) sum of these expressions.
In an ice tray, the water level in any particular ice cube cell will change at a rate proportional to the difference between the cell's water level and the level in the adjacent cells. (a) Argue that a reasonable differential equation model for the water levels x, y, and z in the simplified three-cell tray depicted in the accompanying figure is given by x'=y-x, y'=x+z-2y, z' =y-z. (b) Use eigenvectors to solve the system for the initial conditions x(0) = 6, y(0) = 0, z(0) = 3. (a) The first equation is reasonable because the change in the water level of the first cell, x', is proportional to the difference between the first cell's water level and the level in the adjacent cell, y-x. The constant of proportionality between x' and y-x is k = 1. The third equation is reasonable because the change in the water level of the third cell, z', is proportional to the difference between the third cell's water level and the level in the adjacent cell, y-z. The constant of proportionality between z' and y-x is k = 1. The total rate of change in the water level f the second cell, y', is dependent on the water level in the first cell and the level in the third cell. The difference between the level in the second cell and the level in the first cell is x-y. The difference between the water level in the second cell and the level in the third cell is z-y. Therefore, if y' is the total rate of change in the water level of the second cell, it is reasonable that y' is the (b) Solve the system for the given initial conditions. x(t) y(t) = z(t) sum of these expressions.
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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Subject:calculas
Transcribed Image Text:In an ice tray, the water level in any particular ice cube cell will change at a rate proportional to the difference between the cell's water level and the level in the
adjacent cells.
(a) Argue that a reasonable differential equation model for the water levels x, y, and z in the simplified three-cell tray depicted in the accompanying figure is given
by x'=y-x, y'=x+z-2y, z' =y-z.
(b) Use eigenvectors to solve the system for the initial conditions x(0) = 6, y(0) = 0, z(0) = 3.
(a) The first equation is reasonable because the change in the water level of the first cell, x', is proportional to the difference between the first cell's water level and the level in the adjacent cell,
y-x. The constant of proportionality between x' and y-x is k = 1.
The third equation is reasonable because the change in the water level of the third cell, z', is proportional to the difference between the third cell's water level and the level in the adjacent cell,
y-z. The constant of proportionality between z' and y-x is k = 1.
The total rate of change in the water level f the second cell, y', is dependent on the water level in the first cell and the level in the third cell. The difference between the level in the second cell and
the level in the first cell is x-y.
The difference between the water level in the second cell and the level in the third cell is z-y.
Therefore, if y' is the total rate of change in the water level of the second cell, it is reasonable that y' is the
(b) Solve the system for the given initial conditions.
x(t)
y(t) =
z(t)
sum
of these expressions.
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