29.19.19 problem 532
Internal
problem
ID
[5128]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
19
Problem
number
:
532
Date
solved
:
Sunday, March 30, 2025 at 06:42:45 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: 0.172 (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: 4.831 (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: 0.911 (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)
\[
\log {\left (x \right )} = C_{1} + \log {\left (\frac {\sqrt [3]{1 + \frac {y{\left (x \right )}}{x}}}{\left (- \frac {1}{2} + \frac {y{\left (x \right )}}{x}\right )^{\frac {5}{6}}} \right )}
\]