23.1.385 problem 370

Internal problem ID [4992]
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 : 370
Date solved : Tuesday, September 30, 2025 at 09:13:13 AM
CAS classification : [_rational, _Bernoulli]

\begin{align*} 6 x^{3} y^{\prime }&=4 x^{2} y+\left (1-3 x \right ) y^{4} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 127

ode:=6*x^3*diff(y(x),x) = 4*x^2*y(x)+(1-3*x)*y(x)^4; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {2^{{1}/{3}} \left (-x^{2} \left (-2 c_1 -3 x +\ln \left (x \right )\right )^{2}\right )^{{1}/{3}}}{-2 c_1 -3 x +\ln \left (x \right )} \\ y &= \frac {2^{{1}/{3}} \left (-x^{2} \left (-2 c_1 -3 x +\ln \left (x \right )\right )^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4 c_1 +6 x -2 \ln \left (x \right )} \\ y &= \frac {2^{{1}/{3}} \left (-x^{2} \left (-2 c_1 -3 x +\ln \left (x \right )\right )^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{-4 c_1 -6 x +2 \ln \left (x \right )} \\ \end{align*}

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 99

ode=6*x^3*D[y[x],x]==4*x^2*y[x]+(1-3*x)*y[x]^4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.364 (sec). Leaf size: 88

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*x**3*Derivative(y(x), x) - 4*x**2*y(x) - (1 - 3*x)*y(x)**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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