Solving ODE with Integrating Factor Method: (1-x/y)dx + (2xy+x/y+x^2/y^2)dy = 0

[manenbu]

Homework Statement

Solve:
$$(1-\frac{x}{y})dx + (2xy + \frac{x}{y} + \frac{x^2}{y^2})dy = 0$$

The Attempt at a Solution

No idea what strategy to use here. Tried using an integrating factor, but no success. A lot of x/y in here makes me think I need to use a substitution, but there's also "xy" in here which doesn't help me. What should I do?

 

[TheFurryGoat]

How did you try using the integrating factor? Did you try multiplying by "x^m*y^n" and then solving the integers "m" and "n"?

 

[manenbu]

I tried using the formulas:

$$F(x) = \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N}$$
And then:
$$\mu(x) = e^{\int F(x) dx}$$

(there's also a similar one for y, with signs reversed and M instead of N in the denominator).

Had trouble in the F(x) part - couldn't get a function of x only (or y, for that matter).

 

[manenbu]

Ok! Solved it. Missed a little thing on my side.

Thanks anyway! :)

 

[gabbagabbahey]

The substitution $u(y)=\frac{x}{y}$ works out nicely.

 

[manenbu]

$$C = \ln{|x|} + \ln{|y|} - \frac{x}{y} + y^2$$

That's the solution, in 2 different strategies.
One using the integrating factor, second using your substitution. Thanks for pointing it out. :)