Slope field of dy/dx = -x - y and the solution through (1, 1) Step by Step

Step by step

The question

Question: draw the slope field of dy/dx = −x − y, and the solution through (1, 1).

dydx = −x − y

1 The idea

dy/dx is the slope of a solution curve.

For each point x, y, the equation tells us that slope.

So we draw a short line with that slope at each point: a slope field.

dy/dx = the slope of the solution at (x, y)

slope field: a short line with that slope at each point

2 Slopes at some points

Put x and y of each point into −x − y.

At (0, 0): 0 + 0 = 0.

slope at (0, 0) = 0 + 0 = 0

At (1, 0): −1 + 0 = −1.

slope at (1, 0) = −1 + 0 = −1

slope at (1, 1) = −1 − 1 = −2

Careful!

A point is x first, then y.

Put each one in its own place.

Careful: (x, y): x first, then y

3 Draw the slope lines

Now the grid.

The slopes we found: a short purple line at each point.

Slope 0 is flat, slope 1 goes up like a hill, a minus slope goes down.

The same at every point of the grid.

This is the slope field.

4 Follow the slopes

Start at (1, 1).

Now follow the little lines, like a leaf on a river.

This red curve is the solution through (1, 1).

5 Read the picture

slope 0 where −x − y = 0: y = −x

The lines are flat where −x − y = 0: on the green dashed line, y = −x.

Every solution follows the field, like the red one.

6 Check

Let's check one more point, (2, −1): the slope is −1.

Its line on the grid goes down.

Right!

slope at (2, −1) = −2 − (−1) = −1

Let's look back.

First: the idea.

Then: slopes at some points.

Then: draw the slope lines.

Then: follow the slopes.

Then: read the picture.

Last: check.

Answer