54.8.7 problem 1843

Internal problem ID [13067]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 7, non-linear third and higher order
Problem number : 1843
Date solved : Friday, October 03, 2025 at 03:59:11 AM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 81

ode:=y(x)*diff(diff(diff(y(x),x),x),x)-diff(y(x),x)*diff(diff(y(x),x),x)+y(x)^3*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ -2 \int _{}^{y}\frac {1}{\sqrt {-\textit {\_a}^{4}+4 c_2 \,\textit {\_a}^{2}-4 c_2^{2}+4 c_1}}d \textit {\_a} -x -c_3 &= 0 \\ 2 \int _{}^{y}\frac {1}{\sqrt {-\textit {\_a}^{4}+4 c_2 \,\textit {\_a}^{2}-4 c_2^{2}+4 c_1}}d \textit {\_a} -x -c_3 &= 0 \\ \end{align*}

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - y(x)*Derivative(y(x), (x, 3))/(-y(x)**3 + Derivative(y(x), (x, 2))) cannot be solved by the factorable group method