23.4.256 problem 256

Internal problem ID [6558]
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 : 256
Date solved : Friday, October 03, 2025 at 02:09:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 32

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

\begin{align*} y &= 0 \\ y &= -\frac {x \left (\operatorname {LambertW}\left (-{\mathrm e}^{c_1^{2} c_2 -1} x^{c_1^{2}}\right )+1\right )}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 60.34 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to -\frac {x \left (1+W\left (-e^{-1-c_2 c_1{}^2} x^{c_1{}^2}\right )\right )}{c_1} \end{align*}

✗ Sympy

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

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