7.5.35 problem 35
Internal
problem
ID
[139]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
1.
First
order
differential
equations.
Section
1.6
(substitution
and
exact
equations).
Problems
at
page
72
Problem
number
:
35
Date
solved
:
Friday, February 07, 2025 at 07:57:06 AM
CAS
classification
:
[_exact]
\begin{align*} x^{3}+\frac {y}{x}+\left (y^{2}+\ln \left (x \right )\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 305
dsolve((x^3+y(x)/x)+(y(x)^2+ln(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}-4 \ln \left (x \right )}{2 \left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \left (-\left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}-4 \ln \left (x \right )\right ) \sqrt {3}-\left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (x \right )}{4 \left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \left (\left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (x \right )\right ) \sqrt {3}-\left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (x \right )}{4 \left (-3 x^{4}-12 c_1 +\sqrt {64 \ln \left (x \right )^{3}+9 \left (x^{4}+4 c_1 \right )^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.048 (sec). Leaf size: 326
DSolve[(x^2+y[x]/x)+(y[x]^2+Log[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {-2 \sqrt [3]{2} \log (x)+2^{2/3} \left (-x^3+\sqrt {4 \log ^3(x)+\left (x^3-3 c_1\right ){}^2}+3 c_1\right ){}^{2/3}}{2 \sqrt [3]{-x^3+\sqrt {4 \log ^3(x)+\left (x^3-3 c_1\right ){}^2}+3 c_1}} \\
y(x)\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-x^3+\sqrt {4 \log ^3(x)+\left (x^3-3 c_1\right ){}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) \log (x)}{4 \sqrt [3]{-x^3+\sqrt {4 \log ^3(x)+\left (x^3-3 c_1\right ){}^2}+3 c_1}} \\
y(x)\to \frac {2^{2/3} \left (-1-i \sqrt {3}\right ) \left (-x^3+\sqrt {4 \log ^3(x)+\left (x^3-3 c_1\right ){}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) \log (x)}{4 \sqrt [3]{-x^3+\sqrt {4 \log ^3(x)+\left (x^3-3 c_1\right ){}^2}+3 c_1}} \\
\end{align*}