81.4.22 problem 5-23
Internal
problem
ID
[21536]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
I.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
5.
Integrating
factors.
Page
72.
Problem
number
:
5-23
Date
solved
:
Thursday, October 02, 2025 at 07:47:04 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y \left (x +y+1\right )+x \left (x +3 y+2\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 559
ode:=y(x)*(x+y(x)+1)+x*(x+3*y(x)+2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {{\left (\left (-x^{4}-6 x^{3}+6 \sqrt {6}\, \sqrt {c_1 \left (x^{4}+6 x^{3}+12 x^{2}+54 c_1 +8 x \right )}-12 x^{2}-108 c_1 -8 x \right ) x^{2}\right )}^{{1}/{3}}}{6 x}+\frac {\left (x +2\right )^{2} x}{6 {\left (\left (-x^{4}-6 x^{3}+6 \sqrt {6}\, \sqrt {c_1 \left (x^{4}+6 x^{3}+12 x^{2}+54 c_1 +8 x \right )}-12 x^{2}-108 c_1 -8 x \right ) x^{2}\right )}^{{1}/{3}}}-\frac {x}{6}-\frac {1}{3} \\
y &= \frac {\left (-i \sqrt {3}-1\right ) {\left (-\left (x^{4}+6 x^{3}-6 \sqrt {6}\, \sqrt {c_1 \left (x^{4}+6 x^{3}+12 x^{2}+54 c_1 +8 x \right )}+12 x^{2}+108 c_1 +8 x \right ) x^{2}\right )}^{{2}/{3}}+x \left (x +2\right ) \left (-2 {\left (-\left (x^{4}+6 x^{3}-6 \sqrt {6}\, \sqrt {c_1 \left (x^{4}+6 x^{3}+12 x^{2}+54 c_1 +8 x \right )}+12 x^{2}+108 c_1 +8 x \right ) x^{2}\right )}^{{1}/{3}}+\left (i \sqrt {3}-1\right ) x \left (x +2\right )\right )}{12 {\left (-\left (x^{4}+6 x^{3}-6 \sqrt {6}\, \sqrt {c_1 \left (x^{4}+6 x^{3}+12 x^{2}+54 c_1 +8 x \right )}+12 x^{2}+108 c_1 +8 x \right ) x^{2}\right )}^{{1}/{3}} x} \\
y &= -\frac {\left (1-i \sqrt {3}\right ) {\left (-\left (x^{4}+6 x^{3}-6 \sqrt {3}\, \sqrt {2}\, \sqrt {c_1 \left (x^{4}+6 x^{3}+12 x^{2}+54 c_1 +8 x \right )}+12 x^{2}+108 c_1 +8 x \right ) x^{2}\right )}^{{2}/{3}}+x \left (x +2\right ) \left (2 {\left (-\left (x^{4}+6 x^{3}-6 \sqrt {3}\, \sqrt {2}\, \sqrt {c_1 \left (x^{4}+6 x^{3}+12 x^{2}+54 c_1 +8 x \right )}+12 x^{2}+108 c_1 +8 x \right ) x^{2}\right )}^{{1}/{3}}+\left (1+i \sqrt {3}\right ) x \left (x +2\right )\right )}{12 {\left (-\left (x^{4}+6 x^{3}-6 \sqrt {3}\, \sqrt {2}\, \sqrt {c_1 \left (x^{4}+6 x^{3}+12 x^{2}+54 c_1 +8 x \right )}+12 x^{2}+108 c_1 +8 x \right ) x^{2}\right )}^{{1}/{3}} x} \\
\end{align*}
✓ Mathematica. Time used: 56.951 (sec). Leaf size: 556
ode=y[x]*(x+y[x]+1)+x*(x+3*y[x]+2)*D[y[x],x] ==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {-x^2+\frac {(x+2)^2 x^2}{\sqrt [3]{-x^6-6 x^5-12 x^4-8 x^3+108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 \left (-x^4\right ) \left (x^4+6 x^3+12 x^2+8 x-54 c_1\right )}}}+\sqrt [3]{-x^6-6 x^5-12 x^4-8 x^3+108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 \left (-x^4\right ) \left (x^4+6 x^3+12 x^2+8 x-54 c_1\right )}}-2 x}{6 x}\\ y(x)&\to -\frac {\frac {\left (1+i \sqrt {3}\right ) x^2 (x+2)^2}{\sqrt [3]{-x^6-6 x^5-12 x^4-8 x^3+108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 \left (-x^4\right ) \left (x^4+6 x^3+12 x^2+8 x-54 c_1\right )}}}+\left (1-i \sqrt {3}\right ) \sqrt [3]{-x^6-6 x^5-12 x^4-8 x^3+108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 \left (-x^4\right ) \left (x^4+6 x^3+12 x^2+8 x-54 c_1\right )}}+2 x (x+2)}{12 x}\\ y(x)&\to -\frac {\frac {\left (1-i \sqrt {3}\right ) x^2 (x+2)^2}{\sqrt [3]{-x^6-6 x^5-12 x^4-8 x^3+108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 \left (-x^4\right ) \left (x^4+6 x^3+12 x^2+8 x-54 c_1\right )}}}+\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6-6 x^5-12 x^4-8 x^3+108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 \left (-x^4\right ) \left (x^4+6 x^3+12 x^2+8 x-54 c_1\right )}}+2 x (x+2)}{12 x} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x + 3*y(x) + 2)*Derivative(y(x), x) + (x + y(x) + 1)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out