nonlinear first order DE

utterfly

Hello:
I discovered this forum while looking for advice on solving a first order nonlinear differential equation.
The equation I am trying to solve is

dy/dx=(3ay+3bx^2y^2)/(3x-bx^3y)

a and b are constants. The equation is not exact, nor is it homogeneous. I have failed to separate the variables by factoring. So the usual methods don't work.
Any help or advice will be appreciated.

jeffreydk

Are you sure it isn't homogeneous?

utterfly

Hi Jeffrey
The equation is not homogeneous. See if you can find a work around.
Thanks

tiny-tim

Hi utterfly! Welcome to PF!

(are you sure it isn't (3ax-bx^3y) on the bottom? anyway …)

Hint: first, factor it out as much as you can, then make the obvious substitution.

utterfly

Hello tiny-tim:
It is 3x and not 3ax.
I am going to try an iterative approach. Nothing else seems to work.
Thanks

tiny-tim

Factor it out first!

utterfly

Hi tiny-tim
I have tried factoring the equation - but no luck!
Can you help with the factoring?
Thanks

tiny-tim

dy/dx = (3ay + 3bx²y²)/(3x - bx³y)

= (3y/x)(a + bx²y)/(3 - bx²y)

What's difficult about that?

Now make the obvious substitution …

utterfly

Yes, you can even make it simpler by dividing through by x^2. The terms remaining have both variables. Separation of variables has not been successful.
So substitution will not help.

smallphi

I think he meant to substitute new variable x^2 y(x) = f(x) and then separation of variables x and f works.

tiny-tim

Hi smallphi!

Exactly!

(btw, have you noticed the new x² and x₂ tags on the Reply to thread page? )
Not what I call simpler.
Simplification is always the correct first step.

But obviously it doesn't actually solve the problem.

In hindsight, what part of "Now make the obvious substitution" did you not think worth trying?

Anyway, as smallphi suggests, put z = x²y … what is dz/dx?

arildno

To give you a few further hints:
xy=z/x, and y/x=z/(x^3).

utterfly

I tried the substitution; I do get a result even if looks horrible!
I will repeat the calculation, just to make sure.
Thanks guys!

tiny-tim

Hi utterfly!

It shouldn't look horrible.

What dz/dx did you get?

utterfly

Hi tiny-tim
This is what I get

dz/dx=(z/x)((3a+b)+z(3-2b)/(3-bz))

Solving for z gives a bunch of ln terms.

tiny-tim

hmm … that's not what I get.
If you mean a sum of ln terms, then just antilog the whole thing, and you'll get a neat product of terms, without logs.

utterfly

That is true. Taking anti-logs gives the horrible expression I was referring to.
But, it is a solution!
If you have a simpler expression I would like to see it.
Much appreciate your interest and effort.