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.
The given relation is an implicit solution to the given differential equation.
As, in the given relation , y is defined implicitly as the function of x, so by using implicit differentiation, we will differentiate the given relation concerning x,
Which is identical to the given differential equation.
Thus, the relation is an implicit solution to the differential equation .
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: Let A denote the number of kilograms of salt in the tank at t minutes after the process begins and use the fact that
rate of increase in A =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.]
In Problems 14–24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge–Kutta algorithm. (At the instructor’s discretion, other algorithms may be used.)†
Using the vectorized Runge–Kutta algorithm with h = 0.5, approximate the solution to the initial value problemat t = 8.
Compare this approximation to the actual solution .
Let denote the solution to the initial value problem
⦁ Show that
⦁ Argue that the graph of is decreasing for x near zero and that as x increases from zero, decreases until it crosses the line y = x, where its derivative is zero.
⦁ Let x* be the abscissa of the point where the solution curve crosses the line .Consider the sign of and argue that has a relative minimum at x*.
⦁ What can you say about the graph of for x > x*?
⦁ Verify that y = x – 1 is a solution to and explain why the graph of always stays above the line .
⦁ Sketch the direction field for by using the method of isoclines or a computer software package.
⦁ Sketch the solution using the direction field in part (f).
The motion of a set of particles moving along the x‑axis is governed by the differential equation where denotes the position at time t of the particle.
⦁ If a particle is located at when , what is its velocity at this time?
⦁ Show that the acceleration of a particle is given by
⦁ If a particle is located at when , can it reach the location at any later time?
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