Internal problem ID [10175]
Internal file name [OUTPUT/9122_Monday_June_06_2022_06_42_27_AM_98452448/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
Unable to parse ODE.
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying 5th order ODE linearizable_by_differentiation trying high order reducible trying differential order: 5; mu polynomial in y -> Calling odsolve with the ODE`, diff(diff(diff(diff(diff(y(x), x), x), x), x), x) = (1/9)*(45*(diff(diff(y(x), x), x))*(diff(diff( Integrating factor hint being investigated... trying differential order: 5; exact nonlinear trying differential order: 5; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, diff(diff(diff(_b(_a), _a), _a), _a) = (5/9)*(diff(_b(_a), _a))*(9*(diff(diff(_b(_a), _a), _a))*_b symmetry methods on request `, `high order, trying reduction of order with given symmetries:`[1, 0], [_a, _b]
✓ Solution by Maple
Time used: 0.188 (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 \left (x \right ) &= c_{1} x +c_{2} \\ y \left (x \right ) &= \int \int \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}\frac {1}{\operatorname {RootOf}\left (-20 \ln \left (\textit {\_f} \right )+\int _{}^{\textit {\_Z}}\textit {\_k} \left ({\mathrm e}^{\operatorname {RootOf}\left (81 \textit {\_k}^{2} {\mathrm e}^{\textit {\_Z}}+20 \,{\mathrm e}^{\textit {\_Z}} \ln \left ({\mathrm e}^{\textit {\_Z}}+27\right )-40 \,{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-20 \,{\mathrm e}^{\textit {\_Z}} \ln \left (5\right )+162 c_{1} {\mathrm e}^{\textit {\_Z}}-20 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2187 \textit {\_k}^{2}+540 \ln \left ({\mathrm e}^{\textit {\_Z}}+27\right )-1080 \ln \left (2\right )-540 \ln \left (5\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 d x +c_{4} x +c_{5} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.099 (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]
\[ y(x)\to c_5 x-\frac {4 \sqrt {x (c_3 x+c_2)+c_1}}{c_2{}^2-4 c_1 c_3}+c_4 \]