67.6.3 problem 7.4 (a)

Internal problem ID [16429]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 7. The exact form and general integrating fators. Additional exercises. page 141
Problem number : 7.4 (a)
Date solved : Thursday, October 02, 2025 at 01:31:33 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class B`]]

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

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

ode:=2*x*y(x)+y(x)^2+(2*x*y(x)+x^2)*diff(y(x),x) = 0; 
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: 0.37 (sec). Leaf size: 118

ode=2*x*y[x]+y[x]^2+(2*x*y[x]+x^2)*D[y[x],x]==0; 
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: 0.904 (sec). Leaf size: 36

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

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