Internal problem ID [9432]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1853.
ODE order: 5.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {9 \left (y^{\prime \prime }\right )^{2} y^{\relax (5)}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+40 y^{\prime \prime \prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 118
dsolve(9*diff(diff(y(x),x),x)^2*diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-45*diff(diff(y(x),x),x)*diff(diff(diff(y(x),x),x),x)*diff(diff(diff(diff(y(x),x),x),x),x)+40*diff(diff(diff(y(x),x),x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = c_{1} x +c_{2} \\ y \relax (x ) = \int \left (\int \RootOf \left (-\left (\int _{}^{\textit {\_Z}}\frac {1}{\RootOf \left (-20 \ln \left (\textit {\_f} \right )+\int _{}^{\textit {\_Z}}\textit {\_k} \left ({\mathrm e}^{\RootOf \left (81 \textit {\_k}^{2} {\mathrm e}^{\textit {\_Z}}-40 \,{\mathrm e}^{\textit {\_Z}} \ln \relax (2)-20 \,{\mathrm e}^{\textit {\_Z}} \ln \relax (5)+20 \,{\mathrm e}^{\textit {\_Z}} \ln \left ({\mathrm e}^{\textit {\_Z}}+27\right )+162 c_{1} {\mathrm e}^{\textit {\_Z}}-20 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2187 \textit {\_k}^{2}-1080 \ln \relax (2)-540 \ln \relax (5)+540 \ln \left ({\mathrm e}^{\textit {\_Z}}+27\right )+4374 c_{1}-540 \textit {\_Z} -540\right )}+27\right )d \textit {\_k} +20 c_{2}\right )}d \textit {\_f} \right )+x +c_{3}\right )d x \right )d x +c_{4} x +c_{5} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.107 (sec). Leaf size: 43
DSolve[40*Derivative[3][y][x]^3 - 45*y''[x]*Derivative[3][y][x]*Derivative[4][y][x] + 9*y''[x]^2*Derivative[5][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_5 x-\frac {4 \sqrt {x (c_3 x+c_2)+c_1}}{c_2{}^2-4 c_1 c_3}+c_4 \\ \end{align*}