Internal problem ID [9310]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1732 (book 6.141).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {2 y^{\prime \prime } y-\left (y^{\prime }\right )^{2}-8 y^{3}-4 y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 67
dsolve(2*diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-8*y(x)^3-4*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ \int _{}^{y \relax (x )}\frac {1}{\sqrt {4 \textit {\_a}^{3}+4 \textit {\_a}^{2}+\textit {\_a} c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {1}{\sqrt {4 \textit {\_a}^{3}+4 \textit {\_a}^{2}+\textit {\_a} c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.67 (sec). Leaf size: 351
DSolve[-4*y[x]^2 - 8*y[x]^3 - y'[x]^2 + 2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {i \text {$\#$1} \sqrt {4+\frac {2 c_1}{\text {$\#$1}-\text {$\#$1} \sqrt {1-c_1}}} \sqrt {2+\frac {c_1}{\text {$\#$1}+\text {$\#$1} \sqrt {1-c_1}}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {c_1}{2 \sqrt {1-c_1}+2}}}{\sqrt {\text {$\#$1}}}\right )|\frac {\sqrt {1-c_1}+1}{1-\sqrt {1-c_1}}\right )}{\sqrt {\frac {c_1}{1+\sqrt {1-c_1}}} \sqrt {4 \text {$\#$1}^2+4 \text {$\#$1}+c_1}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {i \text {$\#$1} \sqrt {4+\frac {2 c_1}{\text {$\#$1}-\text {$\#$1} \sqrt {1-c_1}}} \sqrt {2+\frac {c_1}{\text {$\#$1}+\text {$\#$1} \sqrt {1-c_1}}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {c_1}{2 \sqrt {1-c_1}+2}}}{\sqrt {\text {$\#$1}}}\right )|\frac {\sqrt {1-c_1}+1}{1-\sqrt {1-c_1}}\right )}{\sqrt {\frac {c_1}{1+\sqrt {1-c_1}}} \sqrt {4 \text {$\#$1}^2+4 \text {$\#$1}+c_1}}\&\right ][x+c_2] \\ \end{align*}