22.1.31 problem 31
Internal
problem
ID
[4337]
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
:
31
Date
solved
:
Tuesday, September 30, 2025 at 07:20:33 AM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\begin{align*} y+x \left (y^{2}+\ln \left (x \right )\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 233
ode:=y(x)+x*(y(x)^2+ln(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-12 c_1 +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 \ln \left (x \right )}{2 \left (-12 c_1 +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {i \sqrt {3}\, \left (-12 c_1 +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, \ln \left (x \right )+\left (-12 c_1 +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 \ln \left (x \right )}{4 \left (-12 c_1 +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, \left (-12 c_1 +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, \ln \left (x \right )-\left (-12 c_1 +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 \ln \left (x \right )}{4 \left (-12 c_1 +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 0.786 (sec). Leaf size: 272
ode=(y[x])+x*(y[x]^2+Log[x])*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} \log (x)}{\sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}\\ y(x)&\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) \log (x)+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{4 \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) \log (x)}{2^{2/3} \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}}\\ y(x)&\to 0 \end{align*}
✓ Sympy. Time used: 7.798 (sec). Leaf size: 224
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(y(x)**2 + log(x))*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{2} \left (- \frac {\sqrt [3]{2} \left (-1 - \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 \log {\left (x \right )}^{3}}}}{4} + \frac {2 \log {\left (x \right )}}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 \log {\left (x \right )}^{3}}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (- \frac {\sqrt [3]{2} \left (-1 + \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 \log {\left (x \right )}^{3}}}}{4} + \frac {2 \log {\left (x \right )}}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 \log {\left (x \right )}^{3}}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (- \frac {\sqrt [3]{2} \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 \log {\left (x \right )}^{3}}}}{2} + \frac {\log {\left (x \right )}}{\sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + 4 \log {\left (x \right )}^{3}}}}\right )\right ]
\]