$($B$5)$ Verifying Solutions to Differential Equations

Consider the differential equation $y^{\text{'}}y-2x=0.$

Determine whether or not $y=\sqrt[\phantom{2}]{5+2x^2}$ satisfies the differential equation.

$(5+2x^2{)}^{\frac{1}{2}}$

$du=5+2x^2$

$u^{\frac{1}{2}}=\frac{1}{2}{u}^{-\frac{1}{2}}$

$\frac{1}{2}(\sqrt[\phantom{2}]{5+2x^2}{)}^{\frac{1}{2}}(4x)$

Determine whether or not $y=\sqrt[\phantom{2}]{5-2x^2}$ satisfies the differential equation.