32.4.22 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.14, page 90
Internal
problem
ID
[5833]
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.
Exercise
10.14,
page
90
Date
solved
:
Sunday, March 30, 2025 at 10:19:04 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y \left (2 x -y-1\right )+x \left (2 y-x -1\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 385
ode:=y(x)*(2*x-y(x)-1)+x*(2*y(x)-x-1)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {3 \,5^{{1}/{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_1 \,x^{2}+160 c_1 x +80 c_1 -x}{c_1}}-20 x -20\right ) c_1^{2}\right )}^{{1}/{3}}}{40 c_1}+\frac {3 x 5^{{2}/{3}}}{40 {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_1 \,x^{2}+160 c_1 x +80 c_1 -x}{c_1}}-20 x -20\right ) c_1^{2}\right )}^{{1}/{3}}}-x -1 \\
y &= \frac {\frac {3 \left (-i \sqrt {3}-1\right ) 5^{{1}/{3}} \left (-20 c_1^{2} x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right )\right )^{{2}/{3}}}{80}+\frac {3 \left (-\frac {80 \left (x +1\right ) \left (-20 c_1^{2} x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right )\right )^{{1}/{3}}}{3}+\left (i \sqrt {3}-1\right ) x 5^{{2}/{3}}\right ) c_1}{80}}{\left (-20 c_1^{2} x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right )\right )^{{1}/{3}} c_1} \\
y &= \frac {\frac {3 \left (i \sqrt {3}-1\right ) 5^{{1}/{3}} \left (-20 c_1^{2} x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right )\right )^{{2}/{3}}}{80}+\frac {3 \left (-\frac {80 \left (x +1\right ) \left (-20 c_1^{2} x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right )\right )^{{1}/{3}}}{3}+\left (-i \sqrt {3}-1\right ) x 5^{{2}/{3}}\right ) c_1}{80}}{\left (-20 c_1^{2} x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right )\right )^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 37.9 (sec). Leaf size: 471
ode=(y[x]*(2*x-y[x]-1))+(x*(2*y[x]-x-1))*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt [3]{2} x}{\sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}-\frac {\sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}{3 \sqrt [3]{2} c_1}-x-1 \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}-x-1 \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}-x-1 \\
y(x)\to \text {Indeterminate} \\
y(x)\to -x-1 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(-x + 2*y(x) - 1)*Derivative(y(x), x) + (2*x - y(x) - 1)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out