91影视

The direction field for $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{y}$as shown in figure 1.13.

1. Sketch the solution curve that passes through (0, -2). From this sketch, write the equation for the solution.

b. Sketch the solution curve that passes through (-1, 3).

c. What can you say about the solution in part (b) as $\mathbf{x}\mathbf{鈫�}\mathbf{+}\mathbf{鈭�}$? How about $\mathbf{x}\mathbf{鈫�}\mathbf{-}\mathbf{鈭�}$?

(a) Show that ${\mathbf{y}}^{\mathbf{2}}\mathbf{+}\mathbf{x}\mathbf{-}\mathbf{3}\mathbf{=}\mathbf{0}$ is an implicit solution to$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}\mathbf{y}}$ on the interval $\mathbf{(}\mathbf{-}\mathbf{鈭�}\mathbf{,}\mathbf{3}\mathbf{)}$.

(b) Show that${\mathbf{xy}}^{\mathbf{3}}\mathbf{-}{\mathbf{xy}}^{\mathbf{3}}\mathbf{sin}\mathbf{x}\mathbf{=}\mathbf{1}$ is an implicit solution to$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\left(\mathbf{x}\mathbf{cos}\mathbf{x}\mathbf{+}\mathbf{sin}\mathbf{x}\mathbf{-}\mathbf{1}\right)\mathbf{y}}{\mathbf{3}\left(\mathbf{x}\mathbf{-}\mathbf{x}\mathbf{sin}\mathbf{x}\right)}$ on the interval $\left(0,\frac{蟺}{2}\right)$.

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.]

A model for the velocity v at time tof a certain object falling under the influence of gravity in a viscous medium is given by the equation $\frac{\mathbf{dv}}{\mathbf{dt}}\mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{v}}{\mathbf{8}}$.From the direction field shown in Figure 1.14, sketch the solutions with the initial conditions v(0) = 5, 8, and 15. Why is the value v = 8 called the 鈥渢erminal velocity鈥�?

Figure 1.14

The logistic equation for the population (in thousands) of a certain species is given by $\frac{\mathbf{dp}}{\mathbf{dt}}\mathbf{=}\mathbf{3}\mathbf{p}\mathbf{-}\mathbf{2}{\mathbf{p}}^{\mathbf{2}}$.

猞� Sketch the direction field by using either a computer software package or the method of isoclines.

猞� If the initial population is 3000 [that is, p(0) = 3], what can you say about the limiting population?

猞� If $\mathbf{p}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{8}$, what is ${\mathbf{lim}}_{\mathbf{t}鈫�\mathbf{+}鈭�}\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$?

猞� Can a population of 2000 ever decline to 800?

Consider the differential equation $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{x}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{y}$

猞� A solution curve passes through the point $\left(1,\frac{蟺}{2}\right)$. What is its slope at this point?

猞� Argue that every solution curve is increasing for $\mathit{x}\mathbf{>}\mathbf{1}$.

猞� Show that the second derivative of every solution satisfies $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{{\mathbf{dx}}^{\mathbf{2}}}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{x}\mathbf{cos}\mathbf{y}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{sin}\mathbf{2}\mathbf{y}\mathbf{.}$

猞� A solution curve passes through (0,0). Prove that this curve has a relative minimum at (0,0).

The motion of a set of particles moving along the x鈥慳xis is governed by the differential equation $\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{=}{\mathbf{t}}^{\mathbf{3}}\mathbf{-}{\mathbf{x}}^{\mathbf{3}}\mathbf{,}$ where $\mathbf{x}\left(\mathbf{t}\right)$ denotes the position at time t of the particle.

猞� If a particle is located at $\mathit{x}\mathbf{=}\mathbf{1}$ when $\mathit{t}\mathbf{=}\mathbf{1}$ , what is its velocity at this time?

猞� Show that the acceleration of a particle is given by $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{x}}{{\mathbf{dt}}^{\mathbf{2}}}\mathbf{=}\mathbf{3}{\mathbf{t}}^{\mathbf{2}}\mathbf{-}\mathbf{3}{\mathbf{t}}^{\mathbf{3}}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}{\mathbf{x}}^{\mathbf{5}}\mathbf{.}$

猞� If a particle is located at $\mathit{x}\mathbf{=}\mathbf{2}$when $\mathit{t}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{5}$, can it reach the location $\mathit{x}\mathbf{=}\mathbf{1}$at any later time?

[Hint: ${\mathbf{t}}^{\mathbf{3}}\mathbf{-}{\mathbf{x}}^{\mathbf{3}}\mathbf{=}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{x}\mathbf{)}\mathbf{(}{\mathbf{t}}^{\mathbf{2}}\mathbf{+}\mathbf{xt}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}\mathbf{)}\mathbf{.}$]

Question:8. Determine the convergence set of the given power series.