32.4.16 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.8, page 90
Internal
problem
ID
[5827]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
2.
Special
types
of
differential
equations
of
the
first
kind.
Lesson
10
Problem
number
:
Recognizable
Exact
Differential
equations.
Integrating
factors.
Exercise
10.8,
page
90
Date
solved
:
Monday, January 27, 2025 at 01:20:08 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y \left (2 x +y^{3}\right )-x \left (2 x -y^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.153 (sec). Leaf size: 326
dsolve((y(x)*(2*x+y(x)^3))-(x*(2*x-y(x)^3))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\frac {\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}}}{2}+\frac {2 c_{1}^{2}}{\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}}}+c_{1}}{3 x} \\
y \left (x \right ) &= \frac {\left (-1-i \sqrt {3}\right ) \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{2}/{3}}+4 \left (i \sqrt {3}\, c_{1} -c_{1} +\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}}\right ) c_{1}}{12 \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}} x} \\
y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{2}/{3}}+4 \left (-i \sqrt {3}\, c_{1} -c_{1} +\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}}\right ) c_{1}}{12 \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}} x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 15.157 (sec). Leaf size: 371
DSolve[(y[x]*(2*x+y[x]^3))-(x*(2*x-y[x]^3))*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\frac {2 \sqrt [3]{2} c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}+2 c_1}{6 x} \\
y(x)\to \frac {\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}-4 c_1}{12 x} \\
y(x)\to \frac {\frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}-4 c_1}{12 x} \\
y(x)\to 0 \\
\end{align*}