28.1.69 problem 72
Internal
problem
ID
[4375]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
2.
First-Order
and
Simple
Higher-Order
Differential
Equations.
Page
78
Problem
number
:
72
Date
solved
:
Sunday, March 30, 2025 at 03:10:49 AM
CAS
classification
:
[_Bernoulli]
\begin{align*} 3 x y^{\prime }-3 x y^{4} \ln \left (x \right )-y&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 160
ode:=3*x*diff(y(x),x)-3*x*y(x)^4*ln(x)-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {2^{{2}/{3}} {\left (-x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_1 \right )^{2}\right )}^{{1}/{3}}}{6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_1} \\
y &= -\frac {2^{{2}/{3}} {\left (-x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_1 \right )^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{12 x^{2} \ln \left (x \right )-6 x^{2}-8 c_1} \\
y &= \frac {2^{{2}/{3}} {\left (-x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_1 \right )^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{12 x^{2} \ln \left (x \right )-6 x^{2}-8 c_1} \\
\end{align*}
✓ Mathematica. Time used: 0.235 (sec). Leaf size: 120
ode=3*x*D[y[x],x]-3*x*y[x]^4*Log[x]-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: 2.378 (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)
\[
\left [ y{\left (x \right )} = 2^{\frac {2}{3}} \sqrt [3]{- \frac {x}{C_{1} + 6 x^{2} \log {\left (x \right )} - 3 x^{2}}}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{- \frac {x}{C_{1} + 6 x^{2} \log {\left (x \right )} - 3 x^{2}}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{- \frac {x}{C_{1} + 6 x^{2} \log {\left (x \right )} - 3 x^{2}}} \left (-1 + \sqrt {3} i\right )}{2}\right ]
\]