29.19.13 problem 526

Internal problem ID [5122]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 19
Problem number : 526
Date solved : Tuesday, March 04, 2025 at 07:59:27 PM
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: 0.095 (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 \left (x \right ) &= \frac {1+\sqrt {c_{1} x^{2}+1}}{c_{1} x} \\ y \left (x \right ) &= \frac {1-\sqrt {c_{1} x^{2}+1}}{c_{1} x} \\ \end{align*}

✓ Mathematica. Time used: 2.961 (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: 1.948 (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)
 

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