separable first order eqn

**234578** · October 2nd 2010, 01:25 PM

This equation was under the heading "separable equations" and I am supposed to solve it explicitly but I do not see how it is separable since the $x^{2}y$ term cannot be manipulated (in any way I can see) so as to only have functions M(x) and N(y) s.t. $M(x)+N(y)\frac{dy}{dx}=0$

$<br /> \frac{dy}{dx}=\frac{2x}{y+(x^2)y}<br />$
where $y(0)=-2$

Any help would be much appreciated. Can $N(y)=y+x^{2}y$ and $M(x)=-2x$ where x is a constant in N(Y)? but that doesn't make sense because x is not a constant in the equation overall...

**bondesan** · October 2nd 2010, 01:55 PM

$\frac{dy}{dx}=\frac{2x}{y+(x^2)y}\Rightarrow\frac{ dy}{dx}=\frac{2x}{y(1+x^2)}\Rightarrow ydy=\frac{2xdx}{1+x^2}$ . Now integrating both sides will do it.

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