22.1.113 problem 136

Internal problem ID [4419]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 136
Date solved : Tuesday, September 30, 2025 at 07:31:19 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 32

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

\[ y = \frac {{\mathrm e}^{\frac {c_1}{2}} \sqrt {6}}{2 \sqrt {-\frac {{\mathrm e}^{c_1}}{x \operatorname {LambertW}\left (-\frac {2 \,{\mathrm e}^{c_1}}{3 x}\right )}}} \]

✓ Mathematica. Time used: 3.864 (sec). Leaf size: 78

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

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

✓ Sympy. Time used: 5.138 (sec). Leaf size: 114

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

\[ \left [ y{\left (x \right )} = e^{- \frac {C_{1}}{3} - \frac {W\left (\frac {2 \sqrt [3]{-1} e^{- \frac {2 C_{1}}{3}}}{3 x}\right )}{2}}, \ y{\left (x \right )} = e^{- \frac {C_{1}}{3} - \frac {W\left (\frac {\left (- \sqrt [3]{-1} - \left (-1\right )^{\frac {5}{6}} \sqrt {3}\right ) e^{- \frac {2 C_{1}}{3}}}{3 x}\right )}{2}}, \ y{\left (x \right )} = e^{- \frac {C_{1}}{3} - \frac {W\left (\frac {\left (- \sqrt [3]{-1} + \left (-1\right )^{\frac {5}{6}} \sqrt {3}\right ) e^{- \frac {2 C_{1}}{3}}}{3 x}\right )}{2}}\right ] \]