<s>[INST] separable equation1 Does this solve the Impartible Equation is correct?
Impartible Equation:
$(x+1)y'y=y^2-1$
solve:
$y'=y^2-1/y(x+1)$
$f(x,y)=y^2-1/y(x+1)$
$y^2-1/y/(x+1)/1$
$f(x,y)=(y^2-1)/y/(x+1)/1$
$y^2-1=M(x)$
,
$(x+1)/1=N(y)$
 [/INST] The differential equation $(x+1)y'y =y^2-1$ is separable. Rewrite it as $$\frac{y}{y^2-1}y'=\frac{1}{x+1}.$$
On the left we recognize the derivative of $\frac{1}{2}\ln(|y^2-1|$. Integrating, we get
$$\frac{1}{2}\ln(|y^2-1|)=\ln(|x+1|)+C.$$
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