Internal problem ID [9424]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1845.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {9 y^{\prime \prime \prime } y^{2}-45 y^{\prime } y^{\prime \prime } y+40 \left (y^{\prime }\right )^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 85
dsolve(9*y(x)^2*diff(diff(diff(y(x),x),x),x)-45*y(x)*diff(y(x),x)*diff(diff(y(x),x),x)+40*diff(y(x),x)^3=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = {\mathrm e}^{\int \RootOf \left (6 \left (\int _{}^{\textit {\_Z}}\frac {1}{-4 \textit {\_h}^{2}+\sqrt {4 \textit {\_h}^{2} c_{1}+c_{1}^{2}}-c_{1}}d \textit {\_h} \right )+x +c_{2}\right )d x +c_{3}} \\ y \relax (x ) = {\mathrm e}^{\int \RootOf \left (-6 \left (\int _{}^{\textit {\_Z}}\frac {1}{4 \textit {\_h}^{2}+\sqrt {4 \textit {\_h}^{2} c_{1}+c_{1}^{2}}+c_{1}}d \textit {\_h} \right )+x +c_{2}\right )d x +c_{3}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.073 (sec). Leaf size: 21
DSolve[40*y'[x]^3 - 45*y[x]*y'[x]*y''[x] + 9*y[x]^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{(x (c_3 x+c_2)+c_1){}^{3/2}} \\ \end{align*}