Slope field of dy/dx = y - x and the solution through (0, 2) Step by Step

Step by step

The question

Question: draw the slope field of dy/dx = y − x, and the solution through (0, 2).

dydx = y − x

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 y − x.

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

Careful!

x comes first, then y.

At (1, 0), x is 1 and y is 0.

Careful: not at (1, 0): 0 + 1 = 1 ✗

slope at (0, 1) = 0 + 1 = 1

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 (0, 2).

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

This red curve is the solution through (0, 2).

5 Read the picture

slope 0 where y − x = 0: y = x

The lines are flat where y − x = 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 −3.

Its line on the grid goes down, steeply.

Right!

slope at (2, −1) = −2 + (−1) = −3

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