75.5.17 problem 116

Internal problem ID [16682]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 5. Homogeneous equations. Exercises page 44
Problem number : 116
Date solved : Monday, March 31, 2025 at 03:05:20 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 2.9 (sec). Leaf size: 60

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

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

✓ Sympy. Time used: 1.025 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = e^{- \frac {C_{1}}{2} - \frac {W\left (- \frac {e^{- C_{1}}}{x}\right )}{2}} \]