In Problems 25 – 28, use the elimination method to find a general solution for the given system of three equations in the three unknown functions x(t), y(t), z(t).
The solutions for the given linear system are , and .
Elimination Procedure for 2 × 2 Systems:
To find a general solution for the system
Where and are polynomials in :
Let us rewrite the given system of equations into operator form.
Add the equation (4) and (5) together to eliminate z(t),
Multiply D-5 on equation (5). Then, add the equation with equation (6).
Add equations (7) and (8) to get,
Now multiply D-1 on equation (7) and multiply D-2 on equation (8) then add them together.
Since the auxiliary equation to the corresponding homogeneous equation is . The roots are and r=3.
Then, the general solution of y is
Now derivate the equation (10)
Substitute the derivation in equation (9).
Use equations (11) and derivatives of y in equation (5) to get,
So, the solution is founded.
Feedback System with Pooling Delay. Many physical and biological systems involve time delays. A pure time delay has its output the same as its input but shifted in time. A more common type of delay is pooling delay. An example of such a feedback system is shown in Figure 5.3 on page 251. Here the level of fluid in tank B determines the rate at which fluid enters tank A. Suppose this rate is given by where and V are positive constants and is the volume of fluid in tank B at time t.
b. Find a general solution for the system in part (a) when and .
c. Using the general solution obtained in part (b), what can be said about the volume of fluid in each of the tanks as ?
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