Internal problem ID [10072]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1750 (book 6.159).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {4 y^{\prime \prime } y-3 {y^{\prime }}^{2}-12 y^{3}=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 61
dsolve(4*diff(diff(y(x),x),x)*y(x)-3*diff(y(x),x)^2-12*y(x)^3=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{\frac {3}{2}} \left (4 \textit {\_a}^{\frac {3}{2}}+c_{1} \right )}}d \textit {\_a} -x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{\frac {3}{2}} \left (4 \textit {\_a}^{\frac {3}{2}}+c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.963 (sec). Leaf size: 469
DSolve[-12*y[x]^3 - 3*y'[x]^2 + 4*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {4 \text {$\#$1} \sqrt {1+\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {4 \text {$\#$1}^{3/2}}{c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}+c_1\right )}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {4 \text {$\#$1} \sqrt {1+\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {4 \text {$\#$1}^{3/2}}{c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}+c_1\right )}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {4 \text {$\#$1} \sqrt {1-\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {-4 \text {$\#$1}^{3/2}}{-c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}-c_1\right )}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {4 \text {$\#$1} \sqrt {1-\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {-4 \text {$\#$1}^{3/2}}{-c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}-c_1\right )}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {4 \text {$\#$1} \sqrt {1+\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {4 \text {$\#$1}^{3/2}}{c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}+c_1\right )}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {4 \text {$\#$1} \sqrt {1+\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {4 \text {$\#$1}^{3/2}}{c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}+c_1\right )}}\&\right ][x+c_2] \\ \end{align*}