48.1.7 problem Example 3.7
Internal
problem
ID
[7508]
Book
:
THEORY
OF
DIFFERENTIAL
EQUATIONS
IN
ENGINEERING
AND
MECHANICS.
K.T.
CHAU,
CRC
Press.
Boca
Raton,
FL.
2018
Section
:
Chapter
3.
Ordinary
Differential
Equations.
Section
3.2
FIRST
ORDER
ODE.
Page
114
Problem
number
:
Example
3.7
Date
solved
:
Monday, January 27, 2025 at 03:03:05 PM
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: 0.003 (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 {x^{3} \left (i \sqrt {3}-1\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}}} \\
y &= \frac {12 i \sqrt {3}\, x^{3}+i \sqrt {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 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: 4.882 (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*}