Internal problem ID [9369]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1790.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {3 y \left (1-y\right ) y^{\prime \prime }-2 \left (1-2 y\right ) \left (y^{\prime }\right )^{2}-h \relax (y)=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 219
dsolve(3*y(x)*(1-y(x))*diff(diff(y(x),x),x)-2*(1-2*y(x))*diff(y(x),x)^2-h(y(x))=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}-\frac {3}{\sqrt {-6 \textit {\_b}^{2} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}} \left (\int \frac {h \left (\textit {\_b} \right )}{\left (\textit {\_b}^{2}-\textit {\_b} \right )^{\frac {4}{3}} \textit {\_b} \left (\textit {\_b} -1\right )}d \textit {\_b} \right )+9 \textit {\_b}^{2} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}} c_{1}+6 \textit {\_b} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}} \left (\int \frac {h \left (\textit {\_b} \right )}{\left (\textit {\_b}^{2}-\textit {\_b} \right )^{\frac {4}{3}} \textit {\_b} \left (\textit {\_b} -1\right )}d \textit {\_b} \right )-9 \textit {\_b} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}} c_{1}}}d \textit {\_b} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {3}{\sqrt {-6 \textit {\_b}^{2} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}} \left (\int \frac {h \left (\textit {\_b} \right )}{\left (\textit {\_b}^{2}-\textit {\_b} \right )^{\frac {4}{3}} \textit {\_b} \left (\textit {\_b} -1\right )}d \textit {\_b} \right )+9 \textit {\_b}^{2} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}} c_{1}+6 \textit {\_b} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}} \left (\int \frac {h \left (\textit {\_b} \right )}{\left (\textit {\_b}^{2}-\textit {\_b} \right )^{\frac {4}{3}} \textit {\_b} \left (\textit {\_b} -1\right )}d \textit {\_b} \right )-9 \textit {\_b} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}} c_{1}}}d \textit {\_b} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.459 (sec). Leaf size: 186
DSolve[-h[y[x]] - 2*(1 - 2*y[x])*y'[x]^2 + 3*(1 - y[x])*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{(1-K[2])^{2/3} K[2]^{2/3} \sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right ) h(K[1])}{3 (K[1]-1) K[1]}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(1-K[3])^{2/3} K[3]^{2/3} \sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right ) h(K[1])}{3 (K[1]-1) K[1]}dK[1]}}dK[3]\&\right ][x+c_2] \\ \end{align*}