54.2.167 problem 744
Internal
problem
ID
[12041]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
744
Date
solved
:
Wednesday, October 01, 2025 at 12:04:37 AM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {x}{-y+x^{4}+2 x^{2} y^{2}+y^{4}} \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 507
ode:=diff(y(x),x) = x/(-y(x)+x^4+2*x^2*y(x)^2+y(x)^4);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-36 c_1 \,x^{2}-54-c_1^{3}+6 \sqrt {3 x^{2} c_1^{4}+24 x^{4} c_1^{2}+48 x^{6}+3 c_1^{3}+108 c_1 \,x^{2}+81}\right )^{{1}/{3}}}{6}+\frac {c_1^{2}-12 x^{2}}{6 \left (-36 c_1 \,x^{2}-54-c_1^{3}+6 \sqrt {3 x^{2} c_1^{4}+24 x^{4} c_1^{2}+48 x^{6}+3 c_1^{3}+108 c_1 \,x^{2}+81}\right )^{{1}/{3}}}-\frac {c_1}{6} \\
y &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-36 c_1 \,x^{2}-54-c_1^{3}+6 \sqrt {48 x^{6}+24 x^{4} c_1^{2}+\left (3 c_1^{4}+108 c_1 \right ) x^{2}+3 c_1^{3}+81}\right )^{{2}/{3}}+\frac {c_1 \left (-36 c_1 \,x^{2}-54-c_1^{3}+6 \sqrt {48 x^{6}+24 x^{4} c_1^{2}+\left (3 c_1^{4}+108 c_1 \right ) x^{2}+3 c_1^{3}+81}\right )^{{1}/{3}}}{6}+\left (i \sqrt {3}-1\right ) \left (x^{2}-\frac {c_1^{2}}{12}\right )}{\left (-36 c_1 \,x^{2}-54-c_1^{3}+6 \sqrt {48 x^{6}+24 x^{4} c_1^{2}+\left (3 c_1^{4}+108 c_1 \right ) x^{2}+3 c_1^{3}+81}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-36 c_1 \,x^{2}-54-c_1^{3}+6 \sqrt {48 x^{6}+24 x^{4} c_1^{2}+\left (3 c_1^{4}+108 c_1 \right ) x^{2}+3 c_1^{3}+81}\right )^{{2}/{3}}}{12}-\frac {c_1 \left (-36 c_1 \,x^{2}-54-c_1^{3}+6 \sqrt {48 x^{6}+24 x^{4} c_1^{2}+\left (3 c_1^{4}+108 c_1 \right ) x^{2}+3 c_1^{3}+81}\right )^{{1}/{3}}}{6}+\left (1+i \sqrt {3}\right ) \left (x^{2}-\frac {c_1^{2}}{12}\right )}{\left (-36 c_1 \,x^{2}-54-c_1^{3}+6 \sqrt {48 x^{6}+24 x^{4} c_1^{2}+\left (3 c_1^{4}+108 c_1 \right ) x^{2}+3 c_1^{3}+81}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 34.023 (sec). Leaf size: 564
ode=D[y[x],x] == x/(x^4 - y[x] + 2*x^2*y[x]^2 + y[x]^4);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{6 \sqrt [3]{2}}+\frac {2^{2/3} \left (-3 x^2+c_1{}^2\right )}{3 \sqrt [3]{36 c_1 x^2+3 \sqrt {3} \sqrt {16 x^6+32 c_1{}^2 x^4+8 c_1 \left (-9+2 c_1{}^3\right ) x^2+27-8 c_1{}^3}-27+4 c_1{}^3}}+\frac {c_1}{3}\\ y(x)&\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{12 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (3 x^2-c_1{}^2\right )}{3 \sqrt [3]{72 c_1 x^2+6 \sqrt {3} \sqrt {16 x^6+32 c_1{}^2 x^4+8 c_1 \left (-9+2 c_1{}^3\right ) x^2+27-8 c_1{}^3}-54+8 c_1{}^3}}+\frac {c_1}{3}\\ y(x)&\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{12 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (3 x^2-c_1{}^2\right )}{3 \sqrt [3]{72 c_1 x^2+6 \sqrt {3} \sqrt {16 x^6+32 c_1{}^2 x^4+8 c_1 \left (-9+2 c_1{}^3\right ) x^2+27-8 c_1{}^3}-54+8 c_1{}^3}}+\frac {c_1}{3}\\ y(x)&\to -i \sqrt {x^2}\\ y(x)&\to i \sqrt {x^2} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x/(x**4 + 2*x**2*y(x)**2 + y(x)**4 - y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -x/(x**4 + 2*x**2*y(x)**2 + y(x)**4 - y(x)) + Derivative(y(x), x) cannot be solved by the lie group method