Let A, B and C represent three linear differential operators with constant coefficients; for example,
Where the a’s, b’s, and c’s are constants. Verify the following properties:
(a) Commutative laws:
(b) Associative laws:
(c) Distributive law:
Let us prove the commutative property.
Then, find the L.H.S.
.So, A + B = B + A.
Therefore, AB = BA
Henceforth, (A + B) + C = A + (B + C).
Hence, (AB) C = A (BC).
To prove: .
Consequently, A (B + C) = AB + AC.
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 .
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