91Ó°ÊÓ

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.

${\mathbf{e}}^{\mathbf{xy}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{x}\mathbf{-}\mathbf{1}$,$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{xy}}\mathbf{-}\mathbf{y}}{{\mathbf{e}}^{\mathbf{-}\mathbf{xy}}\mathbf{+}\mathbf{x}}$

In Problems 9–20, determine whether the equation is exact.

If it is, then solve it.

${\mathbf{\text{e}}}^{\mathbf{\text{t}}}\left(\text{y - t}\right)\mathbf{\text{dt +}}\left({\text{1 +e}}^{\text{t}}\right)\mathbf{\text{dy = 0}}$

Newton’s law of cooling states that the rate of change in the temperature T(t) of a body is proportional to the difference between the temperature of the medium M(t) and the temperature of the body. That is, $\frac{\mathbf{dT}}{\mathbf{dt}}\mathbf{=}\mathbf{K}\left[\mathbf{M}\left(\mathbf{t}\right)\mathbf{-}\mathbf{T}\left(\mathbf{t}\right)\right]$where K is a constant. Let $\mathbf{K}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{04}{\left(\mathbf{min}\right)}^{\mathbf{-}\mathbf{1}}$and the temperature of the medium be constant, $\mathbf{M}\left(\mathbf{t}\right)\mathbf{=}\mathbf{293}\mathbf{kelvins}$. If the body is initially at 360 kelvins, use Euler’s method with h = 3.0 min to approximate the temperature of the body after

(a) 30 minutes.

(b) 60 minutes.

Verify that $\mathbf{Ï•}\left(x\right)\mathbf{=}\frac{\mathbf{2}}{\left(1-{\mathrm{ce}}^x\right)}\mathbf{,}$where c is an arbitrary constant, it is a one-parameter family of solutions to $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{y}\left(y-2\right)}{\mathbf{2}}$. Graph the solution curves corresponding to $\mathbf{c}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{Â\pm }\mathbf{1}\mathbf{,}\mathbf{Â\pm }\mathbf{2}$ using the same coordinate axes.

In Problems 13-16, write a differential equation that fits the physical description. The rate of change in the temperature T of coffee at time t is proportional to the difference between the temperature M of the air at time t and the temperature of the coffee at time t.

Verify that${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{cy}}^{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{,}$ where c is an arbitrary non-zero constant, is a one-parameter family of implicit solutions to$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{xy}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}$ and graph several of the solution curves using the same coordinate axes.

Show that $\mathbf{Ï•}\left(x\right)\mathbf{=}{\mathbf{Ce}}^{\mathbf{3}\mathbf{x}}\mathbf{+}\mathbf{1}$is a solution tolocalid="1663944867164" $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{3}$for any choice of the constant C. Thus,${\mathbf{Ce}}^{\mathbf{3}\mathbf{x}}\mathbf{+}\mathbf{1}$ is a one-parameter family of solutions to the differential equation. Graph several of the solution curves using the same coordinate axes.

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}{\mathbf{y}}^{\mathbf{4}}\mathbf{-}{\mathbf{x}}^{\mathbf{4}}\mathbf{,}\mathbf{}\mathbf{y}\left(0\right)\mathbf{=}\mathbf{7}$

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{3}\mathbf{x}\mathbf{-}\sqrt[\mathbf{3}]{\mathbf{y}\mathbf{-}\mathbf{1}}\mathbf{,}\mathbf{}\mathbf{y}\left(2\right)\mathbf{=}\mathbf{1}$

(a) For the initial value problem (12) of Example 9. Show that ${\mathbf{Ï•}}_{\mathbf{1}}\left(x\right)\mathbf{=}\mathbf{0}$ and${\mathbf{Ï•}}_{\mathbf{2}}\left(x\right)\mathbf{=}{\left(x-2\right)}^{\mathbf{3}}$are solutions. Hence, this initial value problem has multiple solutions. (See also Project G in Chapter 2.)

(b) Does the initial value problem$\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{3}{\mathit{y}}^{\frac{\mathbf{2}}{\mathbf{3}}}\mathbf{,}$$\mathbf{y}\left(0\right)\mathbf{=}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$have a unique solution in a neighbourhood of$\mathbf{x}\mathbf{=}\mathbf{0}$?