78.5.6 problem 2 (f)
Internal
problem
ID
[18078]
Book
:
DIFFERENTIAL
EQUATIONS
WITH
APPLICATIONS
AND
HISTORICAL
NOTES
by
George
F.
Simmons.
3rd
edition.
2017.
CRC
press,
Boca
Raton
FL.
Section
:
Chapter
2.
First
order
equations.
Section
9
(Integrating
Factors).
Problems
at
page
80
Problem
number
:
2
(f)
Date
solved
:
Monday, March 31, 2025 at 05:07:13 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y+\left (x -2 x^{2} y^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.042 (sec). Leaf size: 316
ode:=y(x)+(x-2*x^2*y(x)^3)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {12^{{1}/{3}} \left (x^{2} 12^{{1}/{3}} c_1 +\left (-9 c_1^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_1^{3}-4 x^{2}}{c_1}}}{9}+c_1 \right ) x^{2}\right )^{{2}/{3}}\right )}{6 c_1 x \left (-9 c_1^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_1^{3}-4 x^{2}}{c_1}}}{9}+c_1 \right ) x^{2}\right )^{{1}/{3}}} \\
y &= \frac {\left (\left (-i \sqrt {3}-1\right ) \left (-9 c_1^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_1^{3}-4 x^{2}}{c_1}}}{9}+c_1 \right ) x^{2}\right )^{{2}/{3}}+\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_1 \,x^{2} 2^{{2}/{3}}\right ) 3^{{1}/{3}} 2^{{2}/{3}}}{12 \left (-9 c_1^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_1^{3}-4 x^{2}}{c_1}}}{9}+c_1 \right ) x^{2}\right )^{{1}/{3}} x c_1} \\
y &= -\frac {\left (\left (1-i \sqrt {3}\right ) \left (-9 c_1^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_1^{3}-4 x^{2}}{c_1}}}{9}+c_1 \right ) x^{2}\right )^{{2}/{3}}+\left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) c_1 \,x^{2} 2^{{2}/{3}}\right ) 3^{{1}/{3}} 2^{{2}/{3}}}{12 \left (-9 c_1^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_1^{3}-4 x^{2}}{c_1}}}{9}+c_1 \right ) x^{2}\right )^{{1}/{3}} x c_1} \\
\end{align*}
✓ Mathematica. Time used: 28.042 (sec). Leaf size: 327
ode=( y[x] )+( x-2*x^2*y[x]^3)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {2 \sqrt [3]{3} c_1 x^2+\sqrt [3]{2} \left (-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}\right ){}^{2/3}}{6^{2/3} x \sqrt [3]{-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}}} \\
y(x)\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (-18 x^2+2 \sqrt {81 x^4-12 c_1{}^3 x^6}\right ){}^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) c_1 x^2}{12 x \sqrt [3]{-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}}} \\
y(x)\to \frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (-18 x^2+2 \sqrt {81 x^4-12 c_1{}^3 x^6}\right ){}^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) c_1 x^2}{12 x \sqrt [3]{-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}}} \\
y(x)\to 0 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-2*x**2*y(x)**3 + x)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out