67.6.15 problem 7.5 (e)

Internal problem ID [16441]
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 : 7.5 (e)
Date solved : Thursday, October 02, 2025 at 01:33:15 PM
CAS classification : [_separable]

\begin{align*} 3 y+3 y^{2}+\left (2 x +4 y x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.165 (sec). Leaf size: 137

ode:=3*y(x)+3*y(x)^2+(2*x+4*x*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-c_1 x -\sqrt {c_1^{2} x^{2}-4 \sqrt {c_1 x}}}{2 c_1 x} \\ y &= \frac {-c_1 x +\sqrt {c_1^{2} x^{2}-4 \sqrt {c_1 x}}}{2 c_1 x} \\ y &= \frac {-c_1 x -\sqrt {c_1^{2} x^{2}+4 \sqrt {c_1 x}}}{2 c_1 x} \\ y &= \frac {-c_1 x +\sqrt {c_1^{2} x^{2}+4 \sqrt {c_1 x}}}{2 c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 6.421 (sec). Leaf size: 67

ode=3*y[x]+3*y[x]^2+(2*x+4*x*y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (-1-\sqrt {1+\frac {4 e^{c_1}}{x^{3/2}}}\right )\\ y(x)&\to \frac {1}{2} \left (-1+\sqrt {1+\frac {4 e^{c_1}}{x^{3/2}}}\right )\\ y(x)&\to -1\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.426 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((4*x*y(x) + 2*x)*Derivative(y(x), x) + 3*y(x)**2 + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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