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Chapter 1: 20E (page 1)
In Problems , solve for , the Laplace transform of the solution to the given initial value problem.
The Initial value forfor
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In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.
,
Use Euler鈥檚 method with step size h = 0.1 to approximate the solution to the initial value problem
, y (1) = 0 at the points .
In Problems 9鈥�20, determine whether the equation is exact.
If it is, then solve it.
In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.
Mixing.Suppose a brine containing 0.2 kg of salt per liter runs into a tank initially filled with 500 L of water containing 5 kg of salt. The brine enters the tank at a rate of 5 L/min. The mixture, kept uniform by stirring, is flowing out at the rate of 5 L/min (see Figure 2.6).
(a)Find the concentration, in kilograms per liter, of salt in the tank after 10 min. [Hint:LetAdenote the number of kilograms of salt in the tank attminutes after the process begins and use the fact that
rate of increase inA=rate of input- rate of exit.
A further discussion of mixing problems is given in Section 3.2.]
(b)After 10 min, a leak develops in the tank and an additional liter per minute of mixture flows out of the tank (see Figure 2.7). What will be the concentration, in kilograms per liter, of salt in the tank 20 min after the leak develops? [Hint:Use the method discussed in Problems 31 and 32.]
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