60.7.106 problem 1718 (book 6.127)

Internal problem ID [11656]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1718 (book 6.127)
Date solved : Sunday, March 30, 2025 at 08:33:37 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime } y+a {y^{\prime }}^{2}+b y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 113

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

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

✓ Mathematica. Time used: 94.317 (sec). Leaf size: 277

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

\begin{align*} \text {Solve}\left [\frac {y(x) \sqrt {(2 a+3) y(x)^{2 a}} \sqrt {1-\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {a+1}{2 a+3},\frac {a+1}{2 a+3}+1,\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}\right )}{(a+1) \sqrt {-2 b y(x)^{2 a+3}+2 a c_1+3 c_1}}&=-x+c_2,y(x)\right ] \\ \text {Solve}\left [\frac {y(x) \sqrt {(2 a+3) y(x)^{2 a}} \sqrt {1-\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {a+1}{2 a+3},\frac {a+1}{2 a+3}+1,\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}\right )}{(a+1) \sqrt {-2 b y(x)^{2 a+3}+2 a c_1+3 c_1}}&=x+c_2,y(x)\right ] \\ \end{align*}

✗ Sympy

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

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