60.8.16 problem 1853

Internal problem ID [11777]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 7, non-linear third and higher order
Problem number : 1853
Date solved : Sunday, March 30, 2025 at 09:15:04 PM
CAS classification : [[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]]

\begin{align*} 9 {y^{\prime \prime }}^{2} y^{\left (5\right )}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+40 y^{\prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.119 (sec). Leaf size: 118

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_1 x +c_2 \\ y &= \int \int \operatorname {RootOf}\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 (5\right )-40 \,{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+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}-540 \ln \left (5\right )-1080 \ln \left (2\right )+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} +x +c_3 \right )d x d x +c_4 x +c_5 \\ \end{align*}

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 43

ode=40*Derivative[3][y][x]^3 - 45*D[y[x],{x,2}]*Derivative[3][y][x]*Derivative[4][y][x] + 9*D[y[x],{x,2}]^2*Derivative[5][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_5 x-\frac {4 \sqrt {x (c_3 x+c_2)+c_1}}{c_2{}^2-4 c_1 c_3}+c_4 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*Derivative(y(x), (x, 2))**2*Derivative(y(x), (x, 5)) - 45*Derivative(y(x), (x, 2))*Derivative(y(x), (x, 3))*Derivative(y(x), (x, 4)) + 40*Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(Dummy_104(x), x) - 9*Dummy_104(x)**2*Derivative(Dummy_104(x), (x, 3))/(5*(9*Dummy_104(x)*Derivative(Dummy_104(x), (x, 2)) - 8)) cannot be solved by the factorable group method