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A homogenous differential equation is an equation of the form

where the two functions, *f* and *g* are both homogenous functions of the same degree. These we can solve by doing the substitution

This means that

or possibly even simpler, by just multiplying through by *dx*, or using the product rule directly

We could multiply our differential equation by *dx* to get

Doing the substitutions above gives us

But since both our functions are homogenous to the same degree we get

where p and q are constants. This gives us

or

and thus

or

By the null factor law, we have

Separating the variables we get

So we have been able to separate the variables. Observe that you are not supposed to follow the above as a recipe but as a proof of concept.

**Example 1**

Let us look at the example from the previous page

We can rewrite this as

Next, we do the substitutions

We get

This gives us

or

Separating the variables gives us

and then we are at the same separable differential equation as in the previous page.

**Example 2**

So both our homogenous functions are of degree 1.

We start by rewriting it as

Next, we do the substitution

This gives us

or

this gives us

or

Separating the variables we get

or

This gives us

Integrating this we get

or

Exponentiation (*e* to the power of) of both sides gives us

or

and finally

It is hard to make y your subject here (you need something called the Lambert W function), but we can make x your subject as in

Then you can reflect that curve through y=x to get the inverse, and thus the curve of the wanted solution. This is what I have done in the graph below.

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