Homework 4
For the following four problems, nd the general solutions of the given di erential equation. If an initial condition is given, nd the corresponding particular
solution. Throughout, primes denote derivatives with respect to x.
1.
2.
3.
4.
5.
y 2y = 3e2x; y(0) = 0
xy + 5y = 7x2 ; y(2) = 5
(1 + x)y + y = cos x; y(0) = 1
xy + (2x 3)y = 4x4
0
0
0
0
A tank initially contains sixty gallons of pure water. Brine containing one
pound of salt per gallon enters the tank at two gallons per minute, and
the (perfectly mixed) solution leaves the tank at three gallons per minute;
thus the tank is empty after exactly one hour.
(a) Find the amount of salt in the tank after t minutes.
(b) What is the maximum amount of salt ever in the tank?
6. Consider a cascade of two tanks shown in Figure 1.5.5, with V1 = 100
gallons and V2 = 200 gallons the volumes of the brine in the two tanks.
Each tank also initially contains fty pounds of salt. The three ow rates
indicated in the gure are each ve gallons per minute, with pure water
owing into tank 1.
(a) Find the amount x(t) of salt in tank 1 at time t.
(b) Suppose that y(t) is the amount of salt in tank 2 at time t. Show
rst that
dy = 5x 5y ;
(1)
dt 100 200
and then solve for y(t) using the function x(t) found in part (a).
(c) Finally, nd the maximum amount of salt ever in tank 2.
7. A 30-year old woman accepts an engineering position with a starting salary
of thirty thousand dollars per year. Her salary S (t) increases exponentially, with S (t) = 30et=20 thousand dollars after t years. meanwhile,
twelve percent of her salary is deposited continuously in a retirement
account, which accumulates interest at a continuous annual rate of six
percent.
(a) Estimate A in terms of t to derive the di erential equation satised by the amount A(t) in her savings account after t years.
(b) Compute A(40), the amount available for her retirement at age 70.
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