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Show that the solution of of the initial value problem $u(0)=1$ of the differential equation
$\left(\frac{d u}{d x}\right)^{2}=\frac{2}{9} \frac{x^{2}}{u^{3}}$
is given by
$u(x)=\left(\frac{5}{6 \sqrt{2}}\left(x^{2}-1\right)+1\right)^{2 / 5}$
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