Homogenous type of first-order differential equations

Let’ start by clarifying what being homogenous type means.

Here is this thing

well, this is a polynomial, but that is not the point.

If we substitute into this,

then voila, will appear in all terms.

This wonderful characteristic is called homogeneity.

This polynomial, for instance, is not homogenous.

This is because if , then x will have a different exponential in each term.

So, that’s it for homogeneity. Now let’s move on to applying this knowledge for solving differential equations.

Let’ solve this one.

The equation is not separable, because if we divided by ,...

then some term will definitely remain on the side.

And that is detrimental to obtaining the solution.

But if we don’t divide, then some y will remain on the side.

Luckily, the degrees are homogenous.

The part has a degree of two...

and so does the part.

substitution, shortly

This equation is now separable, so here comes the separation.

We solve the separable equation, where instead of y, we are going after u.

And when we find u, we turn it back into y.

Let's see another one.

There is such a thing as , this way, bye-bye tangent.

Why don’t we check out another homogeneous equation?

The equation is not separable, but the degrees are homogenous.

It seems the degree is 4.

This is a good sign; we can use the usual substitution.

Now, we get rid of the logarithms.