23.1.256 problem 250

Internal problem ID [4863]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 250
Date solved : Tuesday, September 30, 2025 at 08:45:33 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 115

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

\begin{align*} y &= -\frac {3^{{1}/{3}} \left (x^{2} \left (x^{3}-3 c_1 \right )^{2}\right )^{{1}/{3}}}{x^{3}-3 c_1} \\ y &= \frac {\left (x^{2} \left (x^{3}-3 c_1 \right )^{2}\right )^{{1}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )}{2 x^{3}-6 c_1} \\ y &= \frac {\left (x^{2} \left (x^{3}-3 c_1 \right )^{2}\right )^{{1}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )}{-2 x^{3}+6 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.135 (sec). Leaf size: 89

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

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

✓ Sympy. Time used: 1.162 (sec). Leaf size: 80

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

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