50.1.20 problem 20

Internal problem ID [10018]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 06:46:42 PM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 61

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

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

✓ Mathematica. Time used: 8.39 (sec). Leaf size: 139

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

\begin{align*} y(x)&\to \frac {1}{\sqrt [3]{e^x \left (-3 \int _1^x\frac {1}{3} e^{-K[1]} (1-2 K[1])dK[1]+c_1\right )}}\\ y(x)&\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{e^x \left (-3 \int _1^x\frac {1}{3} e^{-K[1]} (1-2 K[1])dK[1]+c_1\right )}}\\ y(x)&\to \frac {(-1)^{2/3}}{\sqrt [3]{e^x \left (-3 \int _1^x\frac {1}{3} e^{-K[1]} (1-2 K[1])dK[1]+c_1\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.584 (sec). Leaf size: 78

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

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