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**Example 1**

Say we have

So the slopes should be equal to the positions? That ought to lines through the origin. Let us see.

We get

Integrating this gives us

or

Yet again, as in the previous page, the signs from the absolute values can be absorbed into the constant to finally give us

I.e. straight lines through the origin.

**Example 2**

Ok, so the slope is now the negative reciprocal of the position. What could that give us? Let us see. We get

or, after integrating,

Multiplying through by 2 and rearranging a bit gives us

In this we should recognise the equation of a circle. Changing the constant to emphasize that we get

Below we can see a graph of the slope field and a few selected solutions.

It should hopefully make sense that the derivative is the negative reciprocal of the position for these circles that are concentric around the origin. Thing about the radius, tangent property of circles and that the slopes of perpendicular lines are the negative reciprocals of each other’s.

**Example 3**

We get

or

and thus

or

This gives us

Yet again we let the constant absorb the signs. This finally gives us

A slope field with a few particular solutions are shown below.

By the way, as you have seen I usually rename the constant using subscripts as it changes throughout the calculations. The final one is usually written without a subscript.

**Example 4**

If we follow the patterns of the examples we have left to examine

We get

Integration gives us

that gives us

A slope field with a few particular solutions are shown below.

**Yet another example**

This gives us

After integration we get

or

A slope field with a few particular solutions are shown below.

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