Internal problem ID [9422]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1843.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y y^{\prime \prime \prime }-y^{\prime } y^{\prime \prime }+y^{3} y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 81
dsolve(y(x)*diff(diff(diff(y(x),x),x),x)-diff(y(x),x)*diff(diff(y(x),x),x)+y(x)^3*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ \int _{}^{y \relax (x )}-\frac {2}{\sqrt {-\textit {\_a}^{4}+4 \textit {\_a}^{2} c_{2}-4 c_{2}^{2}+4 c_{1}}}d \textit {\_a} -x -c_{3} = 0 \\ \int _{}^{y \relax (x )}\frac {2}{\sqrt {-\textit {\_a}^{4}+4 \textit {\_a}^{2} c_{2}-4 c_{2}^{2}+4 c_{1}}}d \textit {\_a} -x -c_{3} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.43 (sec). Leaf size: 409
DSolve[y[x]^3*y'[x] - y'[x]*y''[x] + y[x]*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {2 i \sqrt {1+\frac {\text {$\#$1}^2}{2 \left (\sqrt {c_2{}^2-c_1}-c_2\right )}} \sqrt {1-\frac {\text {$\#$1}^2}{2 \left (c_2+\sqrt {c_2{}^2-c_1}\right )}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \text {$\#$1}}{\sqrt {2}}\right )|\frac {c_2-\sqrt {c_2{}^2-c_1}}{c_2+\sqrt {c_2{}^2-c_1}}\right )}{\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \sqrt {-\frac {\text {$\#$1}^4}{2}+2 \text {$\#$1}^2 c_2-2 c_1}}\&\right ][x+c_3] \\ y(x)\to \text {InverseFunction}\left [\frac {2 i \sqrt {1+\frac {\text {$\#$1}^2}{2 \left (\sqrt {c_2{}^2-c_1}-c_2\right )}} \sqrt {1-\frac {\text {$\#$1}^2}{2 \left (c_2+\sqrt {c_2{}^2-c_1}\right )}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \text {$\#$1}}{\sqrt {2}}\right )|\frac {c_2-\sqrt {c_2{}^2-c_1}}{c_2+\sqrt {c_2{}^2-c_1}}\right )}{\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \sqrt {-\frac {\text {$\#$1}^4}{2}+2 \text {$\#$1}^2 c_2-2 c_1}}\&\right ][x+c_3] \\ \end{align*}