23.1.545 problem 535
Internal
problem
ID
[5152]
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
:
535
Date
solved
:
Tuesday, September 30, 2025 at 11:46:01 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} x \left (2 x +y\right ) y^{\prime }&=x^{2}+x y-y^{2} \end{align*}
✓ Maple. Time used: 0.183 (sec). Leaf size: 59
ode:=x*(y(x)+2*x)*diff(y(x),x) = x^2+x*y(x)-y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {x \left (\operatorname {RootOf}\left (3 \textit {\_Z}^{15}+\textit {\_Z}^{9}-2 c_1 \,x^{3}\right )^{9}+c_1 \,x^{3}\right )}{-\operatorname {RootOf}\left (3 \textit {\_Z}^{15}+\textit {\_Z}^{9}-2 c_1 \,x^{3}\right )^{9}+2 c_1 \,x^{3}}
\]
✓ Mathematica. Time used: 5.313 (sec). Leaf size: 431
ode=x*(2*x+y[x])*D[y[x],x]==x^2+x*y[x]-y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {$\#$1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\&,1\right ]\\ y(x)&\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {$\#$1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\&,2\right ]\\ y(x)&\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {$\#$1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\&,3\right ]\\ y(x)&\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {$\#$1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\&,4\right ]\\ y(x)&\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {$\#$1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\&,5\right ] \end{align*}
✓ Sympy. Time used: 0.560 (sec). Leaf size: 27
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 + x*(2*x + y(x))*Derivative(y(x), x) - x*y(x) + y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\log {\left (x \right )} = C_{1} + \log {\left (\frac {\sqrt [3]{1 + \frac {y{\left (x \right )}}{x}}}{\left (- \frac {1}{2} + \frac {y{\left (x \right )}}{x}\right )^{\frac {5}{6}}} \right )}
\]