60.7.121 problem 1734 (book 6.143)

Internal problem ID [11671]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1734 (book 6.143)
Date solved : Sunday, March 30, 2025 at 08:41:20 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 78

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

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

✓ Mathematica. Time used: 1.888 (sec). Leaf size: 1353

ode=y[x]^2*(b + a*y[x]) - 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*} \text {Solution too large to show}\end{align*}

✗ Sympy

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