Making a substitution to reach a Chini/Ricatti ODE

Question

I am trying to solve this ODE: $h^2(x)\left(\frac{x^{-2/3}}{3}+h'(x)\right)=-\frac{\pi^2}{2\sqrt{2}}, h(1) = 0$ on $[0,1]$. According to WolframAlpha, this is linked to a Chini/Ricatti equation. Could someone please show me what is the necessary substitution to reach this conclusion? Many thanks

Answer 1

Let $$\eqalign{V(t) &= \int_0^t {\frac {12{s}^{2} \; ds}{3\,{\pi}^{2}\sqrt {2}+4\,{s}^{2}+4\,{s}^{3}}}\cr &= \sum_r \frac{3 r}{3 r + 2} \ln(1-t/r)} $$

where the sum is over the roots of the cubic polynomial $3 \pi^2 \sqrt{2} + 4 x^2 + 4 x^3$.

Then the solution is given implicitly by

$$ V(h(x)/x^{1/3}) + \ln(x) = 0 $$