Suggested languages for you:

Americas

Europe

Q2 E

Expert-verified
Found in: Page 1

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# The direction field for $\frac{\mathbf{dy}}{\mathbf{dx}}{\mathbf{=}}{\mathbf{2}}{\mathbf{x}}{\mathbf{+}}{\mathbf{y}}$ as shown in figure 1.13.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 ${\mathbf{x}}{\mathbf{\to }}{\mathbf{+}}{\mathbf{\infty }}$? How about ${\mathbf{x}}{\mathbf{\to }}{\mathbf{-}}{\mathbf{\infty }}$?

1. The graph is drawn below, and the equation is $\mathrm{y}=-2-2\mathrm{x}$.
2. The graph is drawn below, and the equation is $\mathrm{y}=1-2\mathrm{x}$.
3. The solutions become infinite when $\mathrm{x}\to +\infty \mathrm{or}\mathrm{x}\to -\infty$.
See the step by step solution

## Step 1(a): Find the curve by point (0,-2)

Given $\frac{\mathrm{dy}}{\mathrm{dx}}=2\mathrm{x}+\mathrm{y} ......\left(1\right)$

Put the value of the point (0,-2) in equation (1)

$\mathrm{m}=2\left(0\right)-2=-2$

The curve is $\left(\mathrm{y}+2\right)=-2\left(\mathrm{x}-0\right)$

Hence the solution is ${\mathbf{y}}{\mathbf{=}}{\mathbf{-}}{\mathbf{2}}{\mathbf{-}}{\mathbf{2}}{\mathbf{x}}$.

By putting the different values of x, get the values of y.

## Step 2(b): Find the curve by point (-1,3).

Given $\frac{\mathrm{dy}}{\mathrm{dx}}=2\mathrm{x}+\mathrm{y}. .....\left(2\right)$

Put the value of the point (-1,3) in equation (2)

$\mathrm{m}=2\left(-1\right)-3=1$

The curve is $\mathrm{y}=1-2\mathrm{x}$

Hence the solution is ${\mathbf{y}}{\mathbf{=}}{\mathbf{1}}{\mathbf{-}}{\mathbf{2}}{\mathbf{x}}$.

## Step 3(c): Discuss the solution in part (b) as x→+∞ and x→-∞

As ${\mathbf{x}}{\mathbf{\to }}{\mathbf{\infty }}$ the solution becomes infinite. And when ${\mathbf{x}}{\mathbf{\to }}{\mathbf{-}}{\mathbf{\infty }}$the solution also becomes infinite and has an asymptote ${\mathbf{y}}{\mathbf{=}}{\mathbf{-}}{\mathbf{2}}{\mathbf{-}}{\mathbf{2}}{\mathbf{x}}$.