Slope field of dy/dx = x Easy

The question

Question: draw the slope field of dy/dx = 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.

2 Slopes at some points

Put x and y of each point into x.

At (0, 0): 0 = 0.

At (1, 0): 1 = 1.

Careful!

x comes first, then y.

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

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 Read the picture

The lines are flat where x = 0: on the upright line x = 0.

Every line in a column has the same slope: y does not change it.

Any solution curve follows these lines.

5 Check

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

Its line on the grid goes up, steeply.

Right!

Let's look back.

First: the idea.

Then: slopes at some points.

Then: draw the slope lines.

Then: read the picture.

Last: check.

▶ Watch it solved

Slope field of dy/dx = -x Medium

The question

Question: draw the slope field of dy/dx = −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.

2 Slopes at some points

Put x and y of each point into −x.

At (0, 0): 0 = 0.

At (1, 0): −1 = −1.

Careful!

x comes first, then y.

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

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 Read the picture

The lines are flat where −x = 0: on the upright line x = 0.

Every line in a column has the same slope: y does not change it.

Any solution curve follows these lines.

5 Check

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

Its line on the grid goes down, steeply.

Right!

Let's look back.

First: the idea.

Then: slopes at some points.

Then: draw the slope lines.

Then: read the picture.

Last: check.

▶ Watch it solved

Slope field of dy/dx = xy and the solution through (0, 1) Harder

The question

Question: draw the slope field of dy/dx = xy, and the solution through (0, 1).

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.

2 Slopes at some points

Put x and y of each point into xy.

At (0, 0): 0 × 0 = 0.

At (1, 1): 1 × 1 = 1.

Careful!

A point is x first, then y.

Put each one in its own place.

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, 1).

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

This red curve is the solution through (0, 1).

5 Read the picture

The lines are flat where xy = 0: on the x-axis and on the y-axis.

Every solution follows the field, like the red one.

6 Check

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

Its line on the grid goes down, steeply.

Right!

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.

▶ Watch it solved