Internal problem ID [10059]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1737 (book 6.146).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2}-3 y^{4}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 53
dsolve(2*diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-3*y(x)^4=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (\textit {\_a}^{3}+c_{1} \right )}}d \textit {\_a} -x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (\textit {\_a}^{3}+c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 13.648 (sec). Leaf size: 397
DSolve[-3*y[x]^4 - y'[x]^2 + 2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {\text {$\#$1}^3}{c_1}\right )}{\sqrt {\text {$\#$1}^3+c_1}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {\text {$\#$1}^3}{c_1}\right )}{\sqrt {\text {$\#$1}^3+c_1}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {-\text {$\#$1}^3}{-c_1}\right )}{\sqrt {\text {$\#$1}^3-c_1}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {-\text {$\#$1}^3}{-c_1}\right )}{\sqrt {\text {$\#$1}^3-c_1}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {\text {$\#$1}^3}{c_1}\right )}{\sqrt {\text {$\#$1}^3+c_1}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {\text {$\#$1}^3}{c_1}\right )}{\sqrt {\text {$\#$1}^3+c_1}}\&\right ][x+c_2] \\ \end{align*}