2.2.36 Problem 36

Internal problem ID [10447]
Book : Second order enumerated odes
Section : section 2
Problem number : 36
Date solved : Monday, December 08, 2025 at 08:58:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.2.36.1 ✓ Maple. Time used: 0.016 (sec). Leaf size: 22

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

\[ y = \left (-{\mathrm e}^{c_1} \operatorname {Ei}_{1}\left (-\ln \left (\frac {1}{x}\right )+c_1 \right )+c_2 \right ) x \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
trying 2nd order WeierstrassP 
trying 2nd order JacobiSN 
differential order: 2; trying a linearization to 3rd order 
trying 2nd order ODE linearizable_by_differentiation 
trying 2nd order, 2 integrating factors of the form mu(x,y) 
trying differential order: 2; missing variables 
-> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one \ 
integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 
   --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering meth\ 
ods for dynamical symmetries --- 
   -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through o\ 
ne integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 
trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the sing\ 
ular cases 
trying symmetries linear in x and y(x) 
<- linear symmetries successful
 

2.2.36.2 ✗ Mathematica

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

Not solved

2.2.36.3 ✗ Sympy

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

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