23.1.563 problem 553
Internal
problem
ID
[5170]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
553
Date
solved
:
Tuesday, September 30, 2025 at 11:48:47 AM
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: 0.002 (sec). Leaf size: 385
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 &= \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} \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right ) x \right )^{{2}/{3}}}{80}+\frac {3 \left (-\frac {80 \left (x +1\right ) \left (-20 c_1^{2} \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right ) x \right )^{{1}/{3}}}{3}+\left (i \sqrt {3}-1\right ) 5^{{2}/{3}} x \right ) c_1}{80}}{\left (-20 c_1^{2} \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right ) x \right )^{{1}/{3}} c_1} \\
y &= \frac {\frac {3 \left (i \sqrt {3}-1\right ) 5^{{1}/{3}} \left (-20 c_1^{2} \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right ) x \right )^{{2}/{3}}}{80}+\frac {3 \left (-\frac {80 \left (x +1\right ) \left (-20 c_1^{2} \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right ) x \right )^{{1}/{3}}}{3}+\left (-i \sqrt {3}-1\right ) 5^{{2}/{3}} x \right ) c_1}{80}}{\left (-20 c_1^{2} \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_1 -x}{c_1}}}{20}+x +1\right ) x \right )^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 7.857 (sec). Leaf size: 126
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]
\[
\text {Solve}\left [\int _1^x-\frac {\sqrt [3]{2} 5^{2/3} \left (-(K[2]-1)^3\right )^{2/3}}{27 K[2]-27 K[2]^3}dK[2]+c_1=\int _1^{\frac {(x-1) (5 x+8 y(x)+5)}{2^{2/3} \sqrt [3]{5} \sqrt [3]{-(x-1)^3} (x-2 y(x)+1)}}\frac {1}{K[1]^3+\frac {21 \sqrt [3]{-\frac {1}{2}} K[1]}{2\ 5^{2/3}}+1}dK[1],y(x)\right ]
\]
✗ 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