First order ODE, The Homogeneous Method.

[knowlewj01]

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 can't see a way to do this by separation of variables, is it solvable by integrating factor? anyone got any ideas?

 

[Mark44]

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.

 

[knowlewj01]

ah thanks, i'll give it another go.

 

[Mark44]

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

 

[knowlewj01]

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 can't see how to separate it, can anyone point me in the right direction?

thanks

 

[Mark44]

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.