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20 CHAPTER 1 First-Order Differential Equations

settings. For example, the current in a circuit is often written as a sum of a steady-state term and
a transient term.
The initial ratio of salt to brine in the tank is 100 pounds per 200 gallons or 1/2 pound per
gallon. Since the mixture pumped in has a constant ratio of 1/8 pound per gallon, we expect the
brine mixture to dilute toward the incoming ratio with a terminal amount of salt in the tank of
1/8 pound per gallon times 200 gallons. This leads to the expectation (in the long term) that the
amount of salt in the tank should approach 25, as the model verifies.

SECTION 1.2 PROBLEMS

In each of Problems 1 through 5, find the general solution. dissolved. Beginning at time zero, brine containing 2
pounds of salt per gallon is added at the rate of 3 gal-
3

1. y − y = 2x 2 lons per minute, and the mixture is poured out of the
x
−x
1
x

2. y + y = (e − e ) tank at the rate of 2 gallons per minute. How much salt
2
is in the tank when it contains 100 gallons of brine?

3. y + 2y = x
Hint: The amount of brine in the tank at time t is

4. y + sec(x)y = cos(x) 50 + t.
5. y − 2y =−8x 2

13. Two tanks are connected as in Figure 1.6. Tank 1
In each of Problems 6 through 10, solve the initial value initially contains 20 pounds of salt dissolved in 100
problem. gallons of brine. Tank 2 initially contains 150 gallons
of brine in which 90 pounds of salt are dissolved. At
6. y + 3y = 5e − 6; y(0) = 2 time zero, a brine solution containing 1/2 pound of
2x

7. y + 1 y = 3x; y(3) = 4 salt per gallon is added to tank 1 at the rate of 5 gallons
x−2
per minute. Tank 1 has an output that discharges brine
8. y − y = 2e ; y(0) =−3
4x

into tank 2 at the rate of 5 gallons per minute, and tank

9. y + 2 y = 3; y(0) = 5 2 also has an output of 5 gallons per minute. Deter-
x+1
5y 3 mine the amount of salt in each tank at any time. Also,
10. y + = 3x + x; y(−1) = 4
9x determine when the concentration of salt in tank 2 is a
11. Find all functions with the property that the y intercept minimum and how much salt is in the tank at that time.
2
of the tangent to the graph at (x, y) is 2x . Hint: Solve for the amount of salt in tank 1 at time t
12. A 500 gallon tank initially contains 50 gallons of and use this solution to help determine the amount in
brine solution in which 28 pounds of salt have been tank 2.
5 gal/min; 1/2 lb/gal 5 gal/min
Tank 1 Tank 2
5 gal/min

FIGURE 1.6 Storage tank in Problem 13, Section 1.2.

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October 14, 2010 14:9 THM/NEIL Page-20 27410_01_ch01_p01-42