23.4.228 problem 228

Internal problem ID [6530]
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 : 228
Date solved : Friday, October 03, 2025 at 02:09:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 2 x^{2} y y^{\prime \prime }&=-4 y^{2}+2 x y y^{\prime }+x^{2} {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 25

ode:=2*x^2*y(x)*diff(diff(y(x),x),x) = -4*y(x)^2+2*x*y(x)*diff(y(x),x)+x^2*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {x^{2} \left (2 c_{2} \ln \left (x \right )+c_{1} \right )^{2}}{4 c_{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.259 (sec). Leaf size: 20

ode=2*x^2*y[x]*D[y[x],{x,2}] == -4*y[x]^2 + 2*x*y[x]*D[y[x],x] + x^2*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 x^2 (\log (x)-2 c_1){}^2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*y(x)*Derivative(y(x), (x, 2)) - x**2*Derivative(y(x), x)**2 - 2*x*y(x)*Derivative(y(x), x) + 4*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt((2*x**2*Derivative(y(x), (x, 2)) + 5