I am trying to solve the following:
$$
(x + x^2 + y^2) dy – ydx = 0.
$$
with an integrating factor involving both x and y. Indeed, it seems that an integrating factor of only one variable would not be possible. As $f(x,y) = \frac{y}{x+x^2 + y^2}$ is not homogeneous, it seems that route is closed as well.
My tactic thus far has been to try functions of several forms, such as $x^\alpha y^\beta$ and a few polynomials, but have always come up sort, ending up with higher and higher powers of both $x$ and $y$, meaning that I have not been able to find constants such that any variation brings about an exact DEQ.
I have two questions:
- Is there a general ansatz when dealing with $M$ and $N$ as polynomials of differing degree and with differing constants?
- Can anyone point me in a direction where I might be able to find something like an integrating factor for this?
I'd be most appreciative of any ideas!
Best Answer
Integrating factor is $\mu=\frac{1}{x^2+y^2}$
How i find it ?
Let $\mu=f(t)$, $t=x^2+y^2$, $P=-y$, $Q=x+x^2+y^2$.
Then $\mu'_x=f'(t)2x$, $\mu'_y=f'(t)2y$.
From $(P\mu)'_y=(Q\mu)'_x$ we get differential equation $$tf'(t)+f(t)=0$$ with solution $$f=\frac{C}{t}$$