29.13.13 problem 367
Internal
problem
ID
[4967]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
13
Problem
number
:
367
Date
solved
:
Tuesday, March 04, 2025 at 07:38:58 PM
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: 0.021 (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 \left (x \right ) &= \frac {2^{{1}/{3}} \left (-x^{2} \left (-3 x +\ln \left (x \right )-2 c_{1} \right )^{2}\right )^{{1}/{3}}}{-3 x +\ln \left (x \right )-2 c_{1}} \\
y \left (x \right ) &= \frac {2^{{1}/{3}} \left (-x^{2} \left (-3 x +\ln \left (x \right )-2 c_{1} \right )^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{6 x -2 \ln \left (x \right )+4 c_{1}} \\
y \left (x \right ) &= \frac {2^{{1}/{3}} \left (-x^{2} \left (-3 x +\ln \left (x \right )-2 c_{1} \right )^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{-6 x +2 \ln \left (x \right )-4 c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 0.215 (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: 2.059 (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)
\[
\left [ y{\left (x \right )} = \sqrt [3]{2} \sqrt [3]{\frac {x^{2}}{C_{1} + 3 x - \log {\left (x \right )}}}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{\frac {x^{2}}{C_{1} + 3 x - \log {\left (x \right )}}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{\frac {x^{2}}{C_{1} + 3 x - \log {\left (x \right )}}} \left (-1 + \sqrt {3} i\right )}{2}\right ]
\]