29.23.27 problem 658

Internal problem ID [5249]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 658
Date solved : Tuesday, March 04, 2025 at 08:56:54 PM
CAS classification : [[_homogeneous, `class G`], _exact, _rational, _Bernoulli]

\begin{align*} 6 x y^{2} y^{\prime }+x +2 y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 90

ode:=6*x*y(x)^2*diff(y(x),x)+x+2*y(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= \frac {2^{{1}/{3}} {\left (-\left (x^{2}-4 c_{1} \right ) x^{2}\right )}^{{1}/{3}}}{2 x} \\ y \left (x \right ) &= -\frac {2^{{1}/{3}} {\left (-\left (x^{2}-4 c_{1} \right ) x^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4 x} \\ y \left (x \right ) &= \frac {2^{{1}/{3}} {\left (-\left (x^{2}-4 c_{1} \right ) x^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 x} \\ \end{align*}

✓ Mathematica. Time used: 0.223 (sec). Leaf size: 99

ode=6 x y[x]^2 D[y[x],x]+x+2 y[x]^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{-x^2+4 c_1}}{2^{2/3} \sqrt [3]{x}} \\ y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{-x^2+4 c_1}}{2^{2/3} \sqrt [3]{x}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{-x^2+4 c_1}}{2^{2/3} \sqrt [3]{x}} \\ \end{align*}

✓ Sympy. Time used: 1.398 (sec). Leaf size: 70

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*x*y(x)**2*Derivative(y(x), x) + x + 2*y(x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{\frac {C_{1}}{x} - x}}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (-1 - \sqrt {3} i\right ) \sqrt [3]{\frac {C_{1}}{x} - x}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (-1 + \sqrt {3} i\right ) \sqrt [3]{\frac {C_{1}}{x} - x}}{4}\right ] \]