26.4.5 problem Recognizable Exact Differential equations. Integrating factors. Example 10.741, page 90
Internal
problem
ID
[6951]
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.
Example
10.741,
page
90
Date
solved
:
Tuesday, September 30, 2025 at 04:07:11 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class C`]]
\begin{align*} y^{3}+x y^{2}+y+\left (x^{3}+x^{2} y+x \right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 103
ode:=y(x)^3+x*y(x)^2+y(x)+(x^3+x^2*y(x)+x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \frac {x^{2}+1}{\left (\sqrt {x^{2}+1}\, \sqrt {\frac {-1+\left (x^{4}+x^{2}\right ) c_1}{x^{2} \left (x^{2}+1\right )}}-1\right ) x} \\
y &= \frac {-x^{2}-1}{\left (\sqrt {x^{2}+1}\, \sqrt {\frac {-1+\left (x^{4}+x^{2}\right ) c_1}{x^{2} \left (x^{2}+1\right )}}+1\right ) x} \\
\end{align*}
✓ Mathematica. Time used: 29.81 (sec). Leaf size: 651
ode=(y[x]^3+x*y[x]^2+y[x])+(x^3+x^2*y[x]+x)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {x \left (x^2+1\right )}{-x^2+\frac {\sqrt {\frac {x^3 \exp \left (2 \int _1^x-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{x^2+1}+\left (x^2+1\right ) x \left (-2 \int _1^x-\frac {\exp \left (2 \int _1^{K[2]}-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{K[2]^3+K[2]}dK[2]+c_1\right )}}{\sqrt {\frac {\exp \left (2 \int _1^x-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{x^3+x}}}}\\ y(x)&\to -\frac {x \left (x^2+1\right )}{x^2+\frac {\sqrt {\frac {x^3 \exp \left (2 \int _1^x-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{x^2+1}+\left (x^2+1\right ) x \left (-2 \int _1^x-\frac {\exp \left (2 \int _1^{K[2]}-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{K[2]^3+K[2]}dK[2]+c_1\right )}}{\sqrt {\frac {\exp \left (2 \int _1^x-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{x^3+x}}}}\\ y(x)&\to 0\\ y(x)&\to -\frac {x \left (x^2+1\right ) \sqrt {\frac {\exp \left (2 \int _1^x-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{x^3+x}}}{x^2 \sqrt {\frac {\exp \left (2 \int _1^x-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{x^3+x}}-\sqrt {\frac {x^3 \exp \left (2 \int _1^x-\frac {1}{K[1]^3+K[1]}dK[1]\right )-2 x \left (x^2+1\right )^2 \int _1^x-\frac {\exp \left (2 \int _1^{K[2]}-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{K[2]^3+K[2]}dK[2]}{x^2+1}}}\\ y(x)&\to -\frac {x \left (x^2+1\right ) \sqrt {\frac {\exp \left (2 \int _1^x-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{x^3+x}}}{x^2 \sqrt {\frac {\exp \left (2 \int _1^x-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{x^3+x}}+\sqrt {\frac {x^3 \exp \left (2 \int _1^x-\frac {1}{K[1]^3+K[1]}dK[1]\right )-2 x \left (x^2+1\right )^2 \int _1^x-\frac {\exp \left (2 \int _1^{K[2]}-\frac {1}{K[1]^3+K[1]}dK[1]\right )}{K[2]^3+K[2]}dK[2]}{x^2+1}}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*y(x)**2 + (x**3 + x**2*y(x) + x)*Derivative(y(x), x) + y(x)**3 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out