<s>[INST] Solving a differential equation by separating variables How can one solve the following differential equation by the technique of separation of variables?
$$\frac{1}{x^2}\frac{dy}{dx}=y^5\ \ \  \text{ when }, \ y(0)=-1$$
 [/INST] Notice, $$\frac{1}{x^2}\frac{dy}{dx}=y^5$$
$$\frac{1}{y^2}dy=x^2dx$$
$$\int y^{-5}\ dy=\int x^2\ dx$$
$$\frac{y^{-4}}{-4}=\frac{x^3}{3}+C$$
$$-\frac{1}{4y^{4}}=\frac{x^3}{3}+C$$
now, setting $x=0$ & $y=-1$,
$$-\frac{1}{4(-1)^{4}}=\frac{0}{3}+C\implies C=-\frac{1}{4}$$
set the value of $C$ to get the solution  
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