65.6.14 problem 14

Internal problem ID [15707]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.3.2, page 63
Problem number : 14
Date solved : Thursday, October 02, 2025 at 10:23:37 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 71

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

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

✓ Mathematica. Time used: 0.377 (sec). Leaf size: 118

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

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

✓ Sympy. Time used: 0.918 (sec). Leaf size: 36

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

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