23.4.255 problem 255

Internal problem ID [6557]
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 : 255
Date solved : Friday, October 03, 2025 at 02:09:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 50

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

\begin{align*} y &= 0 \\ y &= \frac {\sqrt {c_1 \left ({\mathrm e}^{-c_2 c_1} x^{c_1}+a \right )}}{c_1} \\ y &= -\frac {\sqrt {c_1 \left ({\mathrm e}^{-c_2 c_1} x^{c_1}+a \right )}}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 14.231 (sec). Leaf size: 77

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

\begin{align*} y(x)&\to -\frac {\sqrt {a+e^{-2 c_1 (-\log (x)+c_2)}}}{\sqrt {2} \sqrt {c_1}}\\ y(x)&\to \frac {\sqrt {a+e^{-2 c_1 (-\log (x)+c_2)}}}{\sqrt {2} \sqrt {c_1}}\\ y(x)&\to \text {Indeterminate} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-a + sqrt(a**2 - 2*a*y(x)**2 + 4*x**2*y(x