In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to 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
a. Make sure that the system is written in operator form.
b. Eliminate one of the variables, say, y, and solve the resulting equation for. If the system is degenerating, stop! A separate analysis is required to determine whether or not there are solutions.
c. (Shortcut) If possible, use the system to derive an equation that involves but not its derivatives. [Otherwise, go to step (d).] Substitute the found expression for into this equation to get a formula for. The expressions for, and give the desired general solution.
d. Eliminate x from the system and solve for. [Solving for gives more constants----in fact, twice as many as needed.]
e. Remove the extra constants by substituting the expressions for and into one or both of the equations in the system. Write the expressions for and in terms of the remaining constants.
Let us rewrite the system in operator form,
Multiply 3 on equation (3) and D-5 on both sides of equation (4) then add equation (3) and (4) together one gets,
Since the auxiliary equation to the corresponding homogeneous equation is:
. The roots are r=7 and r=2.
Then, the homogeneous solution of u is:
Let us take the undetermined coefficients and assume that
Now derivate the equation (7)
Substitute the derivation in equation (5).
Now, equalize the like terms.
Use equations (6) and (8) to get,
Now, take equation (4).
So, the solution is founded.
Two large tanks, each holding 100 L of liquid, are interconnected by pipes, with the liquid flowing from tank A into tank B at a rate of 3 L/min and from B into A at a rate of 1 L/min (see Figure 5.2). The liquid inside each tank is kept well stirred. A brine solution with a concentration of 0.2 kg/L of salt flows into tank A at a rate of 6 L/min. The (diluted) solution flows out of the system from tank A at 4 L/min and from tank B at 2 L/min. If, initially, tank A contains pure water and tank B contains 20 kg of salt, determine the mass of salt in each tank at a time .
Referring to the coupled mass-spring system discussed in Example , suppose an external force is applied to the second object of mass . The displacement functions and now satisfy the system
(a) Show that satisfies the equation
(b) Find a general solution to the equation (18). [Hint: Use undetermined coefficients with .]
(c) Substitute back into (16) to obtain a formula for .
(d) If both masses are displaced 2m to the right of their equilibrium positions and then released, find the displacement functions and .
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