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Unformatted text preview: [SECTION 413! Section 2.3, Problem 2 Problem. A tank initially contains 120 liters of pure water. A mixture
containing a concentration of 7% of Salt enters the tank at a rate of 2%, and the
well—stirred mixture leaves the tank at the same rate. Find an expression in terms
of 7 for the amount of salt in the tank at any time t. Also ﬁnd the limiting amount
ofsaltinthetankastaoo. I ‘ V Solution. One wants a model for the amount M (t) of salt in the tank at a
time t. t should be measured in minutes, and M (t) should be measured in grams,
as those are the given units. I ‘
Entering the tank are 'y[g] of salt, and leaving the tank are 1% M[g]
of salt. So an equation describing the change % of the amount of salt is: dM 2 7;: = +27“ m1” (1” V To solve this equation, which one notes resembles Example 1 in the book, it
may be advantageous to rewrite it in the form 4% + %M = 27. Then one can (as
in Section 2.1) use an integrating factor, Mt) = eéi. Multiplying all terms in the
equation by u(t) turns the equation into eét% + gldeﬁlﬁtM = 276%? By the product
rule, the left side of the equation ‘is now the derivative, with respect to t, of eélﬁtM.
The equation can be rewritten as $(eﬁlﬁtM '= 276th. One can then integrate both sides of the equation with respect to t, turning
the equation into e%tM (t) = 120‘yt'361_oIt + C, for some constant (3'. Then one divides
all terms, including the constant C, by edit and so M (t) = 120 + Ce‘Elﬁt. C' is determined by using the known initial condition, in this case, that the
tank starts with pure water. The amount of salt at time t '= 0 then is zero, or
M(0) = 120 + 061160 = 120 + c = 0. So, 0' = ,—120. The amount of salt in the
tank is M(t) = 120 —120e‘$t. ’ ' The limiting amount of salt in the tank, how much remains as t —> 00, is the
limit of .M (t) = 120 — 120e'6i0t as t grows inﬁnitely large. As t grows, 120 will be
unchanged, but 1206‘6'16t will become inﬁnitessimally small — it will go to Zero. As t ——+ 00, then, the amount of salt in the tank M —> 120[g]. 2.7.39.1} ®= arm/um. a; game (le
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 Spring '04
 Yoon
 Differential Equations, Equations

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