60.7.199 problem 1823 (book 6.232)

Internal problem ID [11749]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1823 (book 6.232)
Date solved : Sunday, March 30, 2025 at 09:11:19 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \left ({y^{\prime }}^{2}+y^{2}\right ) y^{\prime \prime }+y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.387 (sec). Leaf size: 163

ode:=(diff(y(x),x)^2+y(x)^2)*diff(diff(y(x),x),x)+y(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {\sqrt {c_1 +\tan \left (\sqrt {3}\, x \right )}\, {\mathrm e}^{\frac {\sqrt {3}\, \int \frac {\sqrt {\left (9 c_1^{2}+12\right ) \sec \left (\sqrt {3}\, x \right )^{2}+3 c_1^{2}+6 c_1 \tan \left (\sqrt {3}\, x \right )-3}}{c_1 +\tan \left (\sqrt {3}\, x \right )}d x}{6}+c_2}}{\left (\sec \left (\sqrt {3}\, x \right )^{2}\right )^{{1}/{4}}} \\ y &= \frac {\sqrt {c_1 +\tan \left (\sqrt {3}\, x \right )}\, {\mathrm e}^{-\frac {\sqrt {3}\, \int \frac {\sqrt {\left (9 c_1^{2}+12\right ) \sec \left (\sqrt {3}\, x \right )^{2}+3 c_1^{2}+6 c_1 \tan \left (\sqrt {3}\, x \right )-3}}{c_1 +\tan \left (\sqrt {3}\, x \right )}d x}{6}+c_2}}{\left (\sec \left (\sqrt {3}\, x \right )^{2}\right )^{{1}/{4}}} \\ \end{align*}

✓ Mathematica. Time used: 0.678 (sec). Leaf size: 63

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

\[ y(x)\to c_2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]^2+1}{\left (K[1]^2-K[1]+1\right ) \left (K[1]^2+K[1]+1\right )}dK[1]\&\right ][c_1-K[2]]dK[2]\right ) \]

✗ Sympy

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

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