23.4.178 problem 178

Internal problem ID [6480]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 178
Date solved : Tuesday, September 30, 2025 at 03:01:47 PM
CAS classification : [NONE]

\begin{align*} 2 y y^{\prime \prime }&=-1-2 x y^{2}+a y^{3}+{y^{\prime }}^{2} \end{align*}

✗ Maple

ode:=2*y(x)*diff(diff(y(x),x),x) = -1-2*x*y(x)^2+a*y(x)^3+diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=2*y[x]*D[y[x],{x,2}] == -1 - 2*x*y[x]^2 + a*y[x]^3 + D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*y(x)**3 + 2*x*y(x)**2 + 2*y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2 + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-a*y(x)**3 + 2*x*y(x)**2 + 2*y(x)*Derivative(y(x), (x, 2))