Internal problem ID [9362]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1788 (book 6.197).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {2 y \left (1-y\right ) y^{\prime \prime }-\left (1-3 y\right ) {y^{\prime }}^{2}+h \left (y\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 84
dsolve(2*y(x)*(1-y(x))*diff(diff(y(x),x),x)-(1-3*y(x))*diff(y(x),x)^2+h(y(x))=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_b} \left (\int \frac {h \left (\textit {\_b} \right )}{\left (\textit {\_b} -1\right )^{3} \textit {\_b}^{2}}d \textit {\_b} \right )+c_{1} \textit {\_b}}\, \left (\textit {\_b} -1\right )}d \textit {\_b} -x -c_{2} = 0 \\ \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {\textit {\_b} \left (\int \frac {h \left (\textit {\_b} \right )}{\left (\textit {\_b} -1\right )^{3} \textit {\_b}^{2}}d \textit {\_b} \right )+c_{1} \textit {\_b}}\, \left (\textit {\_b} -1\right )}d \textit {\_b} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.614 (sec). Leaf size: 170
DSolve[h[y[x]] - (1 - 3*y[x])*y'[x]^2 + 2*(1 - y[x])*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{(K[2]-1) \sqrt {K[2]} \sqrt {c_1+2 \int _1^{K[2]}\frac {e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )} h(K[1])}{2 (K[1]-1) K[1]}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[3]-1) \sqrt {K[3]} \sqrt {c_1+2 \int _1^{K[3]}\frac {e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )} h(K[1])}{2 (K[1]-1) K[1]}dK[1]}}dK[3]\&\right ][x+c_2] \\ \end{align*}