6.5.33 problem 30

Internal problem ID [1657]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 30
Date solved : Tuesday, September 30, 2025 at 04:41:59 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 82

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

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

✓ Mathematica. Time used: 0.934 (sec). Leaf size: 109

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

\begin{align*} y(x)&\to -x+\sqrt [3]{x^3 (3 \log (x)+1+3 c_1)}\\ y(x)&\to -x+\frac {1}{2} i \left (\sqrt {3}+i\right ) \sqrt [3]{x^3 (3 \log (x)+1+3 c_1)}\\ y(x)&\to -x-\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [3]{x^3 (3 \log (x)+1+3 c_1)} \end{align*}

✗ Sympy

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

TypeError : cannot determine truth value of Relational: -1 < 3*x**3