# First order ODE, The Homogeneous Method.

## Homework Statement

$$\frac{1}{xy}$$ $$\frac{dy}{dx}$$ = $$\frac{1}{(x^2 + 3y^2)}$$

## Homework Equations

used the substitutions:

v = $$\frac{x}{y}$$ ,and
$$\frac{dy}{dx}$$ = $$v + x$$ $$\frac{dv}{dx}$$

## The Attempt at a Solution

took out a factor of xy on the denominator of the term on the right hand side and multiplied through by xy, made the substitutions and at the moment I have something that looks like this:

$$\frac{1}{v + 3/v}$$ = v + x $$\frac{dv}{dx}$$

as it stands I cant see a way to do this by separation of variables, is it solvable by integrating factor? anyone got any ideas?

Mark44
Mentor
You used the substitution v = x/y, but your work in calculating dy/dx looks like you started with y = vx, which is not equivalent to v = x/y.

ah thanks, i'll give it another go.

Mark44
Mentor
I was able to get the equation separated with the substitution v = y/x.

I have the substitutions:

v = $$\frac{y}{x}$$

$$\frac{dy}{dx}$$ = v + x $$\frac{dv}{dx}$$

so now my equation looks a bit like this:

$$\frac{1}{1/v + 3v}$$ = v + x $$\frac{dv}{dx}$$

I played with it a bit and still cant see how to seperate it, can anyone point me in the right direction?

thanks

Mark44
Mentor
What you have is fine, but needs cleaning up using ordinary algebra techniques.

1. Rewrite 1/(1/v + 3v)) as a rational expression - one numerator and one denominator, with no fractions in the top or bottom.
2. Subtract v from both sides, so that you have x*dv/dx on one side, and a rational expression on the other.
3. Divide both sides by x. You should now be able to move dv or dx so that everything in v or dv is on one side, and everything in x or dx is on the other. The equation is now separated, and you can integrate to find the solution.