2.2.36 Problem 36

Internal problem ID [9159]
Book : Second order enumerated odes
Section : section 2
Problem number : 36
Date solved : Sunday, February 23, 2025 at 05:33:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

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

Maple step by step solution

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 methods for dynamical symmetries --- 
   -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) 
trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases 
trying symmetries linear in x and y(x) 
<- linear symmetries successful`
 

Maple dsolve solution

Solving time : 0.036 (sec)
Leaf size : 22

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

\[ y = \left (-{\mathrm e}^{c_{1}} \operatorname {Ei}_{1}\left (-\ln \left (\frac {1}{x}\right )+c_{1} \right )+c_{2} \right ) x \]

✓Mathematica DSolve solution

Solving time : 28.572 (sec)
Leaf size : 33

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

\begin{align*} y(x)\to x \left (e^{c_1} \operatorname {ExpIntegralEi}(-c_1-\log (x))+c_2\right ) \\ y(x)\to c_2 x \\ \end{align*}

✗Sympy solution

Solving time : 0.000 (sec)
Leaf size : 0

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

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