60.1.129 problem 132
Internal
problem
ID
[10143]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
132
Date
solved
:
Wednesday, March 05, 2025 at 08:33:29 AM
CAS
classification
:
[_Bernoulli]
\begin{align*} 3 x y^{\prime }-3 x \ln \left (x \right ) y^{4}-y&=0 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 160
ode:=3*x*diff(y(x),x)-3*x*ln(x)*y(x)^4-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {2^{{2}/{3}} {\left (-x \left (6 \ln \left (x \right ) x^{2}-3 x^{2}-4 c_{1} \right )^{2}\right )}^{{1}/{3}}}{6 \ln \left (x \right ) x^{2}-3 x^{2}-4 c_{1}} \\
y &= -\frac {2^{{2}/{3}} {\left (-x \left (6 \ln \left (x \right ) x^{2}-3 x^{2}-4 c_{1} \right )^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{12 \ln \left (x \right ) x^{2}-6 x^{2}-8 c_{1}} \\
y &= \frac {2^{{2}/{3}} {\left (-x \left (6 \ln \left (x \right ) x^{2}-3 x^{2}-4 c_{1} \right )^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{12 \ln \left (x \right ) x^{2}-6 x^{2}-8 c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 0.24 (sec). Leaf size: 120
ode=3*x*D[y[x],x] - 3*x*Log[x]*y[x]^4 - y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {(-2)^{2/3} \sqrt [3]{x}}{\sqrt [3]{3 x^2-6 x^2 \log (x)+4 c_1}} \\
y(x)\to \frac {2^{2/3} \sqrt [3]{x}}{\sqrt [3]{3 x^2-6 x^2 \log (x)+4 c_1}} \\
y(x)\to -\frac {\sqrt [3]{-1} 2^{2/3} \sqrt [3]{x}}{\sqrt [3]{3 x^2-6 x^2 \log (x)+4 c_1}} \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 2.428 (sec). Leaf size: 109
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-3*x*y(x)**4*log(x) + 3*x*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = 2^{\frac {2}{3}} \sqrt [3]{- \frac {x}{C_{1} + 6 x^{2} \log {\left (x \right )} - 3 x^{2}}}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{- \frac {x}{C_{1} + 6 x^{2} \log {\left (x \right )} - 3 x^{2}}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{- \frac {x}{C_{1} + 6 x^{2} \log {\left (x \right )} - 3 x^{2}}} \left (-1 + \sqrt {3} i\right )}{2}\right ]
\]