23.1.539 problem 529

Internal problem ID [5146]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 529
Date solved : Tuesday, September 30, 2025 at 11:44:35 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.075 (sec). Leaf size: 45

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

\begin{align*} y &= \frac {1+\sqrt {c_1 \,x^{2}+1}}{c_1 x} \\ y &= \frac {1-\sqrt {c_1 \,x^{2}+1}}{c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 1.326 (sec). Leaf size: 80

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

\begin{align*} y(x)&\to \frac {e^{2 c_1}-\sqrt {e^{2 c_1} \left (x^2+e^{2 c_1}\right )}}{x}\\ y(x)&\to \frac {\sqrt {e^{2 c_1} \left (x^2+e^{2 c_1}\right )}+e^{2 c_1}}{x}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.227 (sec). Leaf size: 31

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

\[ \left [ y{\left (x \right )} = \frac {C_{1} - \sqrt {C_{1} \left (C_{1} + x^{2}\right )}}{x}, \ y{\left (x \right )} = \frac {C_{1} + \sqrt {C_{1} \left (C_{1} + x^{2}\right )}}{x}\right ] \]