# $(y^{2} − 1) + 2(x − y(1 + y)^{2})y' = 0$ What method should I use to solve this ODE?

$(y^{2} − 1) + 2(x − y(1 + y)^{2})y' = 0$ What method should I use to solve this ODE ? Clearly This is not linear, and I don't know how to convert this to Bernoulli,finding integration factor seems to be more complicated then the actual question ? Am I missing something ? I think is somewhere along the line $y' = F(y/x)$...but what is $F$

I think I have to do some kind of trig sub...any one can tell me how to transform a 1st Order Ode to polar coordinate ??

-
What about making this ODE exact by multiplying it by a proper integrating factor? – Applied mathematician Nov 19 '12 at 11:21
Ok..how to figure out this integrating factor then ? making $y^2-1 = M$ the coefficient of $y'$ is $N$ $M_y = 2y,N_x = 2$, Then ? – Archer Nov 19 '12 at 11:44

As you did if we put $M(x,y)=y^2-1$ and $N(x,y)=2(x-y\big(1+y\big)^2)$ then $$M_y=2y,N_x=2$$ so according to well-know formula, if the integrating factor be respect to $y$, then $$\mu(y)=\exp\left(\int\frac{M_y-N_x}{-M}dy\right)=\exp\left(\int\frac{2y-2}{1-y^2}dy\right)=\exp\left(\int\frac{-2}{1+y}dy\right)=\exp\left(\ln\left(\frac{1}{(1+y)^2}\right)\right)=\frac{1}{(1+y)^2}$$ Now multiply that to both sides of the equation. It makes your equation exact.