29.20.5 problem 550
Internal
problem
ID
[5146]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
20
Problem
number
:
550
Date
solved
:
Tuesday, March 04, 2025 at 08:18:03 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} x \left (1+x -2 y\right ) y^{\prime }+\left (1-2 x +y\right ) y&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 387
ode:=x*(1+x-2*y(x))*diff(y(x),x)+(1-2*x+y(x))*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \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 \left (x \right ) &= \frac {\frac {3 \,5^{{1}/{3}} \left (-1-i \sqrt {3}\right ) {\left (-20 x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2}\right )}^{{2}/{3}}}{80}+\frac {3 c_{1} \left (\frac {80 \left (-x -1\right ) {\left (-20 x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2}\right )}^{{1}/{3}}}{3}+x \left (i \sqrt {3}-1\right ) 5^{{2}/{3}}\right )}{80}}{{\left (-20 x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
y \left (x \right ) &= \frac {\frac {3 \left (i \sqrt {3}-1\right ) 5^{{1}/{3}} {\left (-20 x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2}\right )}^{{2}/{3}}}{80}+\frac {3 \left (-\frac {80 \left (x +1\right ) {\left (-20 x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2}\right )}^{{1}/{3}}}{3}+\left (-1-i \sqrt {3}\right ) x 5^{{2}/{3}}\right ) c_{1}}{80}}{{\left (-20 x \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 41.087 (sec). Leaf size: 471
ode=x(1+x-2 y[x])D[y[x],x]+(1-2 x+y[x])y[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