Exact Equations & Substitutions · Challenge problem

The Riccati equation $y' = y^2 - \dfrac{2}{x^2}$ has the particular solution $y_1 = \dfrac1x$. Use the substitution $y = \dfrac1x + u$ to solve it with $y(1) = 2$ for $x > 0$, and find $y\!\left(\dfrac12\right)$.