26.3.3 problem Exact Differential equations. Exercise 9.6, page 79

Internal problem ID [6936]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 9
Problem number : Exact Differential equations. Exercise 9.6, page 79
Date solved : Tuesday, September 30, 2025 at 04:07:00 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 205

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

\begin{align*} y &= -\frac {2 \left (c_1 \,x^{2}-\frac {\left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{2}/{3}}}{4}\right )}{\left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{1}/{3}}}{4 \sqrt {c_1}}-\frac {\sqrt {c_1}\, \left (i \sqrt {3}-1\right ) x^{2}}{\left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{1}/{3}}} \\ y &= \frac {4 i \sqrt {3}\, c_1 \,x^{2}+i \sqrt {3}\, \left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{2}/{3}}+4 c_1 \,x^{2}-\left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{2}/{3}}}{4 \left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\ \end{align*}

✓ Mathematica. Time used: 0.097 (sec). Leaf size: 42

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]^2+1}{K[1] \left (K[1]^2+3\right )}dK[1]=-\log (x)+c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out