81.1.41 problem 2-39
Internal
problem
ID
[21486]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
I.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
2.
Separable
differential
equations
Problem
number
:
2-39
Date
solved
:
Thursday, October 02, 2025 at 07:41:42 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y+x y^{\prime }+\frac {y^{3} \left (y-x y^{\prime }\right )}{x}&=0 \end{align*}
✓ Maple. Time used: 0.295 (sec). Leaf size: 287
ode:=y(x)+x*diff(y(x),x)+y(x)^3/x*(y(x)-x*diff(y(x),x)) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {3^{{1}/{3}} \left (x^{2} 3^{{1}/{3}} c_1 +{\left (-9 x \left (-\frac {\sqrt {3}\, \sqrt {\frac {-x^{4}+27 c_1^{3}}{c_1}}}{9}+c_1 \right ) c_1^{2}\right )}^{{2}/{3}}\right )}{3 c_1 {\left (-9 x \left (-\frac {\sqrt {3}\, \sqrt {\frac {-x^{4}+27 c_1^{3}}{c_1}}}{9}+c_1 \right ) c_1^{2}\right )}^{{1}/{3}}} \\
y &= \frac {\left (\left (-i \sqrt {3}-1\right ) {\left (-9 x \left (-\frac {\sqrt {3}\, \sqrt {\frac {-x^{4}+27 c_1^{3}}{c_1}}}{9}+c_1 \right ) c_1^{2}\right )}^{{2}/{3}}+x^{2} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_1 \right ) 3^{{1}/{3}}}{6 {\left (-9 x \left (-\frac {\sqrt {3}\, \sqrt {\frac {-x^{4}+27 c_1^{3}}{c_1}}}{9}+c_1 \right ) c_1^{2}\right )}^{{1}/{3}} c_1} \\
y &= -\frac {3^{{1}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (-9 x \left (-\frac {\sqrt {3}\, \sqrt {\frac {-x^{4}+27 c_1^{3}}{c_1}}}{9}+c_1 \right ) c_1^{2}\right )}^{{2}/{3}}+x^{2} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) c_1 \right )}{6 {\left (-9 x \left (-\frac {\sqrt {3}\, \sqrt {\frac {-x^{4}+27 c_1^{3}}{c_1}}}{9}+c_1 \right ) c_1^{2}\right )}^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 59.435 (sec). Leaf size: 514
ode=(y[x]+x*D[y[x],x])+y[x]^3/x*(y[x]-x*D[y[x],x]) ==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {4 x^2}{\sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 e^{8 c_1} x^2-64 e^{4 c_1} x^6}-9 e^{4 c_1} x}}+\frac {e^{-\frac {4 c_1}{3}} \sqrt [3]{\sqrt {3} \sqrt {27 e^{8 c_1} x^2-64 e^{4 c_1} x^6}-9 e^{4 c_1} x}}{3^{2/3}}\\ y(x)&\to -\frac {2 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 e^{8 c_1} x^2-64 e^{4 c_1} x^6}-9 e^{4 c_1} x}}-\frac {\left (1-i \sqrt {3}\right ) e^{-\frac {4 c_1}{3}} \sqrt [3]{\sqrt {3} \sqrt {27 e^{8 c_1} x^2-64 e^{4 c_1} x^6}-9 e^{4 c_1} x}}{2\ 3^{2/3}}\\ y(x)&\to -\frac {2 \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 e^{8 c_1} x^2-64 e^{4 c_1} x^6}-9 e^{4 c_1} x}}-\frac {\left (1+i \sqrt {3}\right ) e^{-\frac {4 c_1}{3}} \sqrt [3]{\sqrt {3} \sqrt {27 e^{8 c_1} x^2-64 e^{4 c_1} x^6}-9 e^{4 c_1} x}}{2\ 3^{2/3}}\\ y(x)&\to \sqrt [3]{\sqrt {x^2}-x}\\ y(x)&\to -\frac {\left (\sqrt {3}-3 i\right ) \sqrt [3]{\sqrt {x^2}-x}}{2 \sqrt {3}}\\ y(x)&\to \frac {\sqrt [3]{\sqrt {x^2}-x} \text {Root}\left [\text {$\#$1}^6-192\&,3\right ]}{2 \sqrt [6]{3}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), x) + y(x) + (-x*Derivative(y(x), x) + y(x))*y(x)**3/x,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out