Step by step
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.