60.2.37 problem Problem 52

Internal problem ID [15217]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 52
Date solved : Thursday, October 02, 2025 at 10:07:25 AM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} -y y^{\prime }-{y^{\prime }}^{2} x +x y y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 17

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

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\frac {c_1 \,x^{2}}{2}} c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 19

ode=x*y[x]*D[y[x],{x,2}]-x*D[y[x],x]^2-y[x]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt((4*x**2*Derivative(y(x), (x, 2)) + y(x))*y(x)) - y(x))/(2*x) cannot be solved by the factorable group method