50.6.6 problem 1(f)
Internal
problem
ID
[7898]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
1.
What
is
a
differential
equation.
Section
1.8.
Integrating
Factors.
Page
32
Problem
number
:
1(f)
Date
solved
:
Wednesday, March 05, 2025 at 05:16:54 AM
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: 0.226 (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 (12^{{1}/{3}} c_{1} x^{2}+\left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{{2}/{3}}\right )}{6 c_{1} x \left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{{1}/{3}}} \\
y &= \frac {2^{{2}/{3}} \left (\left (-i \sqrt {3}-1\right ) \left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{{2}/{3}}+2^{{2}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) x^{2} c_{1} \right ) 3^{{1}/{3}}}{12 \left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{{1}/{3}} x c_{1}} \\
y &= -\frac {2^{{2}/{3}} \left (\left (1-i \sqrt {3}\right ) \left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{{2}/{3}}+2^{{2}/{3}} x^{2} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) c_{1} \right ) 3^{{1}/{3}}}{12 \left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{{1}/{3}} x c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 30.078 (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