88.5.7 problem 7
Internal
problem
ID
[23981]
Book
:
Elementary
Differential
Equations.
By
Lee
Roy
Wilcox
and
Herbert
J.
Curtis.
1961
first
edition.
International
texbook
company.
Scranton,
Penn.
USA.
CAT
number
61-15976
Section
:
Chapter
2.
Differential
equations
of
first
order.
Exercise
at
page
35
Problem
number
:
7
Date
solved
:
Thursday, October 02, 2025 at 09:48:54 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\begin{align*} 2 y \ln \left (x \right ) \ln \left (y\right )+x \left (\ln \left (x \right )^{2}+\ln \left (y\right )^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 228
ode:=2*y(x)*ln(x)*ln(y(x))+x*(ln(x)^2+ln(y(x))^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= {\mathrm e}^{\frac {\left (12 c_1 +4 \sqrt {4 \ln \left (x \right )^{6}+9 c_1^{2}}\right )^{{1}/{3}}}{2}-\frac {2 \ln \left (x \right )^{2}}{\left (12 c_1 +4 \sqrt {4 \ln \left (x \right )^{6}+9 c_1^{2}}\right )^{{1}/{3}}}} \\
y &= {\mathrm e}^{-\frac {\left (1+i \sqrt {3}\right ) \left (12 c_1 +4 \sqrt {4 \ln \left (x \right )^{6}+9 c_1^{2}}\right )^{{1}/{3}}}{4}-\frac {\ln \left (x \right )^{2} \left (i \sqrt {3}-1\right )}{\left (12 c_1 +4 \sqrt {4 \ln \left (x \right )^{6}+9 c_1^{2}}\right )^{{1}/{3}}}} \\
y &= {\mathrm e}^{\frac {i \sqrt {3}\, \left (12 c_1 +4 \sqrt {4 \ln \left (x \right )^{6}+9 c_1^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, \ln \left (x \right )^{2}-\left (12 c_1 +4 \sqrt {4 \ln \left (x \right )^{6}+9 c_1^{2}}\right )^{{2}/{3}}+4 \ln \left (x \right )^{2}}{4 \left (12 c_1 +4 \sqrt {4 \ln \left (x \right )^{6}+9 c_1^{2}}\right )^{{1}/{3}}}} \\
\end{align*}
✓ Mathematica. Time used: 0.087 (sec). Leaf size: 567
ode=( 2*y[x]*Log[x] *Log[y[x]] ) + x*( Log[x]^2 +Log[y[x]]^2 )*D[y[x],{x,1}]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \fbox {$\exp \left (\frac {\sqrt [3]{3 c_1+\sqrt {4 \log ^6(x)+9 c_1{}^2}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} \log ^2(x)}{\sqrt [3]{3 c_1+\sqrt {4 \log ^6(x)+9 c_1{}^2}}}\right )\text { if }\text {condition}$}\\ y(x)&\to \fbox {$\exp \left (\frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) \log ^2(x)+i 2^{2/3} \left (i+\sqrt {3}\right ) \left (3 c_1+\sqrt {4 \log ^6(x)+9 c_1{}^2}\right ){}^{2/3}}{4 \sqrt [3]{3 c_1+\sqrt {4 \log ^6(x)+9 c_1{}^2}}}\right )\text { if }\text {condition}$}\\ y(x)&\to \fbox {$\exp \left (\frac {\left (1-i \sqrt {3}\right ) \log ^2(x)}{2^{2/3} \sqrt [3]{3 c_1+\sqrt {4 \log ^6(x)+9 c_1{}^2}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{3 c_1+\sqrt {4 \log ^6(x)+9 c_1{}^2}}}{2 \sqrt [3]{2}}\right )\text { if }\text {condition}$} \end{align*}
✓ Sympy. Time used: 47.074 (sec). Leaf size: 192
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(log(x)**2 + log(y(x))**2)*Derivative(y(x), x) + 2*y(x)*log(x)*log(y(x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = e^{- \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + \log {\left (x \right )}^{6}}}}{2} + \frac {2 \log {\left (x \right )}^{2}}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + \log {\left (x \right )}^{6}}}}}, \ y{\left (x \right )} = e^{- \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + \log {\left (x \right )}^{6}}}}{2} + \frac {2 \log {\left (x \right )}^{2}}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + \log {\left (x \right )}^{6}}}}}, \ y{\left (x \right )} = e^{- \sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + \log {\left (x \right )}^{6}}} + \frac {\log {\left (x \right )}^{2}}{\sqrt [3]{3 C_{1} + \sqrt {9 C_{1}^{2} + \log {\left (x \right )}^{6}}}}}\right ]
\]