54.7.118 problem 1731 (book 6.140)

Internal problem ID [12967]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1731 (book 6.140)
Date solved : Wednesday, October 01, 2025 at 02:55:26 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y^{\prime \prime } y-{y^{\prime }}^{2}-8 y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 57

ode:=2*diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-8*y(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.553 (sec). Leaf size: 415

ode=-8*y[x]^3 - D[y[x],x]^2 + 2*y[x]*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {4 \text {$\#$1}^2}{c_1}\right )}{\sqrt {4 \text {$\#$1}^2+c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {4 \text {$\#$1}^2}{c_1}\right )}{\sqrt {4 \text {$\#$1}^2+c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {-4 \text {$\#$1}^2}{-c_1}\right )}{\sqrt {4 \text {$\#$1}^2-c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {-4 \text {$\#$1}^2}{-c_1}\right )}{\sqrt {4 \text {$\#$1}^2-c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {4 \text {$\#$1}^2}{c_1}\right )}{\sqrt {4 \text {$\#$1}^2+c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {4 \text {$\#$1}^2}{c_1}\right )}{\sqrt {4 \text {$\#$1}^2+c_1}}\&\right ][x+c_2] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*y(x)**3 + 2*y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(2)*sqrt((-4*y(x)**2 + Derivative(y(x), (x, 2)))*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method