54.7.30 problem 1620 (6.30)

Internal problem ID [12879]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1620 (6.30)
Date solved : Wednesday, October 01, 2025 at 02:43:53 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 332

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

\begin{align*} 2 \int _{}^{y}\frac {\left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{1}/{3}}}{\textit {\_a}^{4}-\textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{1}/{3}}+\left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{2}/{3}}}d \textit {\_a} -x -c_2 &= 0 \\ -4 \int _{}^{y}\frac {\left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{1}/{3}}}{-i \sqrt {3}\, \textit {\_a}^{4}+i \left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{2}/{3}} \sqrt {3}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{1}/{3}}+\left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{2}/{3}}}d \textit {\_a} -x -c_2 &= 0 \\ 4 \int _{}^{y}-\frac {\left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{1}/{3}}}{i \sqrt {3}\, \textit {\_a}^{4}-i \left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{2}/{3}} \sqrt {3}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{1}/{3}}+\left (\textit {\_a}^{6}+2 c_1 +2 \sqrt {c_1 \left (\textit {\_a}^{6}+c_1 \right )}\right )^{{2}/{3}}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✗ Mathematica

ode=-y[x]^3 + y[x]*D[y[x],x] + D[y[x],{x,2}] == 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 + y(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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