<s>[INST] Solve $y'=\frac{2}{x^2}-y^{2}$ 
$$y'=\frac{2}{x^2}-y^{2}$$
  $y=\frac{k}{x}$ is a solution, find the general solution 

So we have riccati differential equation.
$y=z+\frac{k}{x}\iff z=y-\frac{k}{x}$
$y'=z'-\frac{k}{x^2}$
Plugin it to the ode we get:
$$z'-\frac{k}{x^2}=\frac{2}{x^2}-(z+\frac{k}{x})^2$$
$$z'=\frac{2-k^2+k}{x^2}-\frac{2kz}{x}-z^2$$
Which is again a riccati differential equation do I need to guess a particular $k$? for example $k=-1$?
 [/INST] Hint: if $y=\frac{k}{x}$ is a solution then  pluging in the equation we get  $$\\ -\frac { k }{ { x }^{ 2 } } =\frac { 2 }{ { x }^{ 2 } } -\frac { { k }^{ 2 } }{ { x }^{ 2 } } \\ { k }^{ 2 }-k-2=0\\ \left( k-2 \right) \left( k+1 \right) =0\\ $$
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