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**Example 1**

Let us solve

This is a separable equation, but let us first solve it as a linear differential equation. We have *p*(*x*)= –*x* so that gives us

Multiplying our equation by this gives us

or

So

If we now do the variable substitution

We get

Multiplying away the first factor, we get

that is our final solution. Interestingly enough it is the same solution as we found to

but just shifted down one unit. As we can see we could write

as

or

so as mentioned it is separable. We get

or

and thus

and finally

just as expected.

**Example 2**

Let us take an old example and modify it slightly. If the slopes were directed out from the origin we had that

and that gave us solutions of the form

I.e. straight lines through the origin. Now let us change that a bit. Say we are subtracting or adding) a bit from the slope, and that what we subtract is proportional to the distance from the *y*-axis. Say we have

This can be written as

So it is a linear first-order equation with

This gives us

Multiplying through with this gives us

or

Integration now gives us

that gives us

This looks rather pleasing as a graph, with a few particular solutions.

As you can see all the solutions will be parabolas passing through the origin.

To find particular solutions passing through a particular point (*x*_{0},*y*_{0}) we just substitute in those values into our solution to get

or

This gives us the particular solution

Link to Geogebra file on this.

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