60.2.172 problem 748
Internal
problem
ID
[10746]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
748
Date
solved
:
Sunday, March 30, 2025 at 06:33:08 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {y \left (x +y\right )}{x \left (x +y^{3}\right )} \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 312
ode:=diff(y(x),x) = y(x)*(x+y(x))/x/(x+y(x)^3);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (27 x +3 \sqrt {-24 c_1^{3}-72 c_1^{2} \ln \left (x \right )-72 c_1 \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{2}/{3}}+6 \ln \left (x \right )+6 c_1}{3 \left (27 x +3 \sqrt {-24 c_1^{3}-72 c_1^{2} \ln \left (x \right )-72 c_1 \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (27 x +3 \sqrt {-24 c_1^{3}-72 c_1^{2} \ln \left (x \right )-72 c_1 \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{2}/{3}}}{6}+\left (i \sqrt {3}-1\right ) \left (\ln \left (x \right )+c_1 \right )}{\left (27 x +3 \sqrt {-24 c_1^{3}-72 c_1^{2} \ln \left (x \right )-72 c_1 \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (27 x +3 \sqrt {-24 c_1^{3}-72 c_1^{2} \ln \left (x \right )-72 c_1 \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{2}/{3}}}{6}+\left (-i \sqrt {3}-1\right ) \left (\ln \left (x \right )+c_1 \right )}{\left (27 x +3 \sqrt {-24 c_1^{3}-72 c_1^{2} \ln \left (x \right )-72 c_1 \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 9.632 (sec). Leaf size: 300
ode=D[y[x],x] == (y[x]*(x + y[x]))/(x*(x + y[x]^3));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {2 \sqrt [3]{2} (\log (x)+1+c_1)}{\sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+1+c_1){}^3}}}+\frac {\sqrt [3]{9 x+\frac {1}{6} \sqrt {2916 x^2-864 (\log (x)+1+c_1){}^3}}}{3^{2/3}} \\
y(x)\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+1+c_1){}^3}}}{6 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}\right ) (\log (x)+1+c_1)}{\sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+1+c_1){}^3}}} \\
y(x)\to -\frac {\sqrt [3]{2} \left (1-i \sqrt {3}\right ) (\log (x)+1+c_1)}{\sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+1+c_1){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+1+c_1){}^3}}}{6 \sqrt [3]{2}} \\
y(x)\to 0 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - (x + y(x))*y(x)/(x*(x + y(x)**3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (x + y(x))*y(x)/(x*(x + y(x)**3)) cannot be solved by the lie group method