76.4.25 problem 31
Internal
problem
ID
[17419]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
2.
First
order
differential
equations.
Section
2.6
(Exact
equations
and
integrating
factors).
Problems
at
page
100
Problem
number
:
31
Date
solved
:
Tuesday, January 28, 2025 at 10:06:35 AM
CAS
classification
:
[_rational]
\begin{align*} 3 x +\frac {6}{y}+\left (\frac {x^{2}}{y}+\frac {3 y}{x}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 322
dsolve((3*x+6/y(x))+ (x^2/y(x)+3*y(x)/x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {-12 x^{3}+\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{{2}/{3}}}{6 \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{{1}/{3}}}{12}-\frac {\left (i \sqrt {3}-1\right ) x^{3}}{\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{{1}/{3}}} \\
y &= \frac {12 i \sqrt {3}\, x^{3}+i \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{{2}/{3}} \sqrt {3}+12 x^{3}-\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{{2}/{3}}}{12 \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 4.838 (sec). Leaf size: 331
DSolve[(3*x+6/y[x]) + (x^2/y[x]+3*y[x]/x)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} x^3}{\sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}} \\
y(x)\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) x^3}{2^{2/3} \sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^3}{2^{2/3} \sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}} \\
\end{align*}