54.7.124 problem 1737 (book 6.146)

Internal problem ID [12973]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1737 (book 6.146)
Date solved : Wednesday, October 01, 2025 at 02:55:30 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y^{\prime \prime } y-{y^{\prime }}^{2}-3 y^{4}&=0 \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 53

ode:=2*diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-3*y(x)^4 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 4.511 (sec). Leaf size: 397

ode=-3*y[x]^4 - 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 {$\#$1}} \sqrt {1+\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {\text {$\#$1}^3}{c_1}\right )}{\sqrt {\text {$\#$1}^3+c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {\text {$\#$1}^3}{c_1}\right )}{\sqrt {\text {$\#$1}^3+c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {-\text {$\#$1}^3}{-c_1}\right )}{\sqrt {\text {$\#$1}^3-c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {-\text {$\#$1}^3}{-c_1}\right )}{\sqrt {\text {$\#$1}^3-c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {\text {$\#$1}^3}{c_1}\right )}{\sqrt {\text {$\#$1}^3+c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {\text {$\#$1}^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {\text {$\#$1}^3}{c_1}\right )}{\sqrt {\text {$\#$1}^3+c_1}}\&\right ][x+c_2] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*y(x)**4 + 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((-3*y(x)**3 + 2*Derivative(y(x), (x, 2)))*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method