23.1.555 problem 545

Internal problem ID [5162]
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 : 545
Date solved : Tuesday, September 30, 2025 at 11:47:07 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _Bernoulli]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 47

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

\begin{align*} y &= -\frac {\sqrt {3}\, \sqrt {-x \left (x^{3}-3 c_1 \right )}}{3 x} \\ y &= \frac {\sqrt {3}\, \sqrt {-x \left (x^{3}-3 c_1 \right )}}{3 x} \\ \end{align*}

✓ Mathematica. Time used: 0.135 (sec). Leaf size: 60

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

\begin{align*} y(x)&\to -\frac {\sqrt {-x^3+3 c_1}}{\sqrt {3} \sqrt {x}}\\ y(x)&\to \frac {\sqrt {-x^3+3 c_1}}{\sqrt {3} \sqrt {x}} \end{align*}

✓ Sympy. Time used: 0.298 (sec). Leaf size: 39

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {3} \sqrt {\frac {C_{1}}{x} - x^{2}}}{3}, \ y{\left (x \right )} = \frac {\sqrt {3} \sqrt {\frac {C_{1}}{x} - x^{2}}}{3}\right ] \]