In Problems $3-6,y=\frac{1}{x^2+c}$ is a one$-$parameter family of solutions of the first$-$order $DE{y}^{\text{'}}+2x{y}^2=0.$ Find a solution of the first$-$order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined.

$3.y(2)=\frac{1}{3}$

$4.y(-2)=\frac{1}{2}$

$5.y(0)=1$

$6.y(\frac{1}{2})=-4$