Chapter 2: First-Order Differential Equations

Each DE in Problemsis a Bernoulli equation. In Problemssolve the given differential equation by using an appropriate substitution.

(a) Identify the nullclines (see Problem 17) in Problems 1,3,and 4. With a colored pencil, circle any lineal elements in Figures 2.1.12, 2.1.14, and 2.1.15 that you think maybe a lineal element at a point on a nullcline.

(b) What are the nullclines of an autonomous first-order DE?

In Problems 1–22, solve the given differential equation by separation of variables.

$\frac{\text{dN}}{\text{dt}}{\text{+ N = N te}}^{\text{t + 2}}$

Classify each differential equation as separable, exact, linear, homogeneous, or Bernoulli. Some equations may be more than one kind. Do not solve.

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

In Problems, 1-20 determine whether the given differential equation is exact. If it is exact, solve it.

$\mathbf{(}\mathbf{4}{\mathit{t}}^{\mathbf{3}}\mathit{y}\mathbf{-}\mathbf{15}{\mathit{t}}^{\mathbf{2}}\mathbf{-}\mathit{y}\mathbf{)}\mathit{d}\mathit{t}\mathbf{+}\mathbf{(}{\mathit{t}}^{\mathbf{4}}\mathbf{+}\mathbf{3}{\mathit{y}}^{\mathbf{2}}\mathbf{-}\mathit{t}\mathbf{)}\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}$

Each DE in Problemsis a Bernoulli equation. In Problemssolve the given differential equation by using an appropriate substitution.

Consider the autonomous first-order differential equationand the initial condition. By hand, sketch the graph of a typical solution when y0has the given values.

(a) (b)

(c) (d)

In problems 1-24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution.

$(x+1)\frac{dy}{dx}+(x+2)y=2x{e}^{-x}$

In Problems 19-26 solve the given differential equation.

$\mathbf{(}{\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}\mathit{d}\mathit{x}\mathbf{=}\mathit{y}\mathit{s}\mathit{e}{\mathit{c}}^{\mathbf{2}}\mathit{x}\mathit{d}\mathit{y}$

n Problems, 1–4 reproduces the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve.

$$\begin{gathered} \frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{-}{\mathit{y}}^{\mathbf{2}} \\ \mathit{a}\mathbf{\left)}\mathit{y}\mathbf{\right(}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{=}\mathbf{1} \\ \mathit{b}\mathbf{)}\mathit{y}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{=}\mathbf{0} \\ \mathit{c}\mathbf{\left)}\mathit{y}\mathbf{\right(}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{2} \\ \mathit{d}\mathbf{)}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0} \end{gathered}$$

FIGURE 2.1.12 Direction field for Problem1