Chapter 2: First-Order Differential Equations

In Problems 1 and 2 use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of (3) by hand, first using $\mathit{h}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}$and then using$\mathit{h}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$.

1.${\mathit{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{2}\mathit{x}\mathbf{-}\mathbf{3}\mathit{y}\mathbf{+}\mathbf{1}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{5}\mathbf{;}\mathbf{}\mathit{y}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{)}$

Each DE in Problems $1-14$is homogeneous. In Problems localid="1654933062855" $1-10$solve the given differential equation by using an appropriate substitutionlocalid="1654933173507" $(x-y)dx+xdy=0$

Find the general solution of the given differential equation. Give the largest interval / over which the general solution is defined. Determine whether there are any transient terms in the general solution.

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{5}\mathbf{y}$

In 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.

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

FIGURE 2.1.12 Direction field for Problem 1

In problems, determine whether the given differential equation is exact. If it is exact, solve it.

$(2x-1)dx+(3y+7)dy=0$

Answer Problems 1-12 without referring back to the text. Fill in the blanks or answer true or false.

The linear DE, ${\mathit{y}}^{\cancel{\mathbf{c}}}\mathbf{-}\mathit{k}\mathit{y}\mathbf{=}\mathit{A}$, whereandare constants, is autonomous. The critical point of the equation is a(n) (attractor or repeller) for k > 0anda(n) (attractor or repeller) for k < 0.

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

(a) (b)

(c) (d)

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

Use the method discussed under “Equations of the Form $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{G}\left(\mathbf{ax}\mathbf{+}\mathbf{by}\right)$” to solve problems 17-20. $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{sin}\left(\mathbf{x}\mathbf{-}\mathbf{y}\right)$

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

$\left(\frac{1}{t}+\frac{1}{t^2}-\frac{y}{t^2+y^2}\right)\mathit{d}\mathit{t}\mathbf{+}\left(y{e}^y+\frac{t}{t^2+y^2}\right)\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}$