29.20.6 problem 551
Internal
problem
ID
[5147]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
20
Problem
number
:
551
Date
solved
:
Sunday, March 30, 2025 at 06:45:14 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: 386
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 (1+i \sqrt {3}\right ) 5^{{1}/{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (-1+x \right )^{2} c_1 -x}{c_1}}+20 x -20\right ) c_1^{2}\right )}^{{2}/{3}}}{80}+\frac {3 c_1 \left (\frac {80 \left (-1+x \right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (-1+x \right )^{2} c_1 -x}{c_1}}+20 x -20\right ) c_1^{2}\right )}^{{1}/{3}}}{3}+x 5^{{2}/{3}} \left (i \sqrt {3}-1\right )\right )}{80}}{c_1 {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (-1+x \right )^{2} c_1 -x}{c_1}}+20 x -20\right ) c_1^{2}\right )}^{{1}/{3}}} \\
y &= \frac {\frac {3 \left (i \sqrt {3}-1\right ) 5^{{1}/{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (-1+x \right )^{2} c_1 -x}{c_1}}+20 x -20\right ) c_1^{2}\right )}^{{2}/{3}}}{80}+\frac {3 \left (-\frac {80 \left (1-x \right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (-1+x \right )^{2} c_1 -x}{c_1}}+20 x -20\right ) c_1^{2}\right )}^{{1}/{3}}}{3}+\left (-i \sqrt {3}-1\right ) x 5^{{2}/{3}}\right ) c_1}{80}}{c_1 {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (-1+x \right )^{2} c_1 -x}{c_1}}+20 x -20\right ) c_1^{2}\right )}^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 40.386 (sec). Leaf size: 463
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 {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}+\frac {\sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+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 {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+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 {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+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