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Say we have an equation of the form

or, equivalently,

This we will solve by doing a variable substitution. Let

That means that

This gives us that our original equation will become

so

and

Now we can integrate both sides separately. In practice, it will often end up way easier than what it looks like from the above.

**Example 1 **

We have already solved equations of this form, but let us try to solve one of them using this method to see that we get the same result. Let us look at

The substitutions above give us

This gives us

or

We can now do the substitution

to get

or, after integrations

or

and thus

or

or, given that *y*=*vx*

Then we multiply through by *x*^{2} to get

I.e. the same solution as before, but in a way harder way. So let us look at an example where this method helps.

**Example 2**

Say we have

We can rewrite this as

So it is an equation of the desired form.

Next, we do the substitutions, and I suggest you do it on the original equation. We get

or

or,

By the null factor law either *x*=0 or

This gives us

or

The LHS we can solve by the substitution

to get

Integration gives us

or

and

This gives us

and

or, given that *y*=*vx*

or

Multiplying through by *x*^{4} gives us

A graph of this is shown below.

Interestingly enough the slope field is scale-invariant (as you could figure out from how the equation is defined).

We will look at this example again later on.

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