60.8.9 problem 1845

Internal problem ID [11770]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 7, non-linear third and higher order
Problem number : 1845
Date solved : Sunday, March 30, 2025 at 09:14:40 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]]

\begin{align*} 9 y^{2} y^{\prime \prime \prime }-45 y y^{\prime } y^{\prime \prime }+40 {y^{\prime }}^{3}&=0 \end{align*}

✓ Maple. Time used: 0.121 (sec). Leaf size: 85

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

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (-6 \int _{}^{\textit {\_Z}}\frac {1}{4 \textit {\_h}^{2}+\sqrt {c_1 \left (4 \textit {\_h}^{2}+c_1 \right )}+c_1}d \textit {\_h} +x +c_2 \right )d x +c_3} \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (6 \int _{}^{\textit {\_Z}}-\frac {1}{4 \textit {\_h}^{2}-\sqrt {c_1 \left (4 \textit {\_h}^{2}+c_1 \right )}+c_1}d \textit {\_h} +x +c_2 \right )d x +c_3} \\ \end{align*}

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 21

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

\[ y(x)\to \frac {1}{(x (c_3 x+c_2)+c_1){}^{3/2}} \]

✗ Sympy

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

NotImplementedError : The given ODE (sqrt(59049*y(x)**4*Derivative(y(x), (x, 3))**2/1600 - 19683*y(x)**3*Derivative(y(x), (x, 2))**3/128)/2 + 243*y(x)**2*Derivative(y(x), (x, 3))/80)**(1/3)/3 + Derivative(y(x), x) + 9*y(x)*Derivative(y(x), (x, 2))/(8*(sqrt(59049*y(x)**4*Derivative(y(x), (x, 3))**2/1600 - 19683*y(x)**3*Derivative(y(x), (x, 2))**3/128)/2 + 243*y(x)**2*Derivative(y(x), (x, 3))/80)**(1/3)) cannot be solved by the factorable group method