91影视

In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.

$\mathbf{6}\mathbf{-}{\mathit{t}}^{\mathbf{3}}\left(\frac{d^{2}v}{d{t}^2}\right)\mathbf{+}\mathbf{3}\mathit{v}\mathbf{-}\mathit{l}\mathit{n}\mathit{t}\mathbf{=}\frac{\mathbf{d}\mathbf{v}}{\mathbf{d}\mathbf{t}}$

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).

In Problems 3-8, determine whether the given function is a solution to the given differential equation.

$\mathbf{x}\mathbf{=}\mathbf{cos}\mathbf{}\mathbf{2}\mathbf{t}$,$\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{+}\mathbf{tx}\mathbf{=}\mathbf{sin}\mathbf{}\mathbf{2}\mathbf{t}$

In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.

$\mathit{x}\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\mathbf{3}\mathit{x}\mathbf{-}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}{\mathit{e}}^{\mathbf{3}\mathbf{x}}$

Consider the differential equation $\frac{\mathbf{dp}}{\mathbf{dt}}\mathbf{=}\mathbf{p}\mathbf{(}\mathbf{p}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{2}\mathbf{-}\mathbf{p}\mathbf{)}$ for the population p (in thousands) of a certain species at time t.

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

猞� If the initial population is 4000 [that is, $p\left(\mathbf{0}\right)=\mathbf{4}$], what can you say about the limiting population ${\mathbf{lim}}_{\mathbf{t}鈫�鈭�}\mathbf{P}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{?}$

猞� If $\mathit{p}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{7}$, what is ${\mathbf{lim}}_{\mathbf{t}鈫�鈭�}\mathbf{P}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{?}$

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

猞� Can a population of 900 ever increase to 1100?

In Problems 3鈥�8, determine whether the given function is a solution to the given differential equation.

$\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\mathbf{-}\mathbf{3}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}$,$\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{{\mathbf{dx}}^{\mathbf{2}}}\mathbf{-}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{0}$

Decide whether the statement made is True or False. The function $\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{-}\mathbf{3}}\mathbf{sin}\mathbf{t}\mathbf{+}\mathbf{4}{\mathbf{t}}^{\mathbf{-}\mathbf{3}}$ is a solution to ${\mathbf{t}}^{\mathbf{3}}\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{=}\mathbf{cos}\mathbf{t}\mathbf{-}\mathbf{3}{\mathbf{t}}^{\mathbf{2}}\mathbf{x}$.

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.

Question: In Problems 3鈥�8, determine whether the given function is a solution to the given differential equation.

$\mathbf{y}\mathbf{=}\mathbf{3}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{2}\mathbf{x}\mathbf{+}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}$, $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{5}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}$