54.2.210 problem 787
Internal
problem
ID
[12084]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
787
Date
solved
:
Wednesday, October 01, 2025 at 12:14:03 AM
CAS
classification
:
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class B`]]
\begin{align*} y^{\prime }&=\frac {x \left (-x -1+x^{2}-2 x^{2} y+2 x^{4}\right )}{\left (x^{2}-y\right ) \left (x +1\right )} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 191
ode:=diff(y(x),x) = 1/(x^2-y(x))*x*(-x-1+x^2-2*x^2*y(x)+2*x^4)/(1+x);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {4 x^{2} {\mathrm e}^{\operatorname {RootOf}\left (8 x^{3} {\mathrm e}^{\textit {\_Z}}-24 \,{\mathrm e}^{\textit {\_Z}} x^{2}-36 x^{3}+6 \ln \left (\frac {2 \,{\mathrm e}^{\textit {\_Z}}-9}{\left (x +1\right )^{4}}\right ) {\mathrm e}^{\textit {\_Z}}+18 c_1 \,{\mathrm e}^{\textit {\_Z}}-6 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+24 x \,{\mathrm e}^{\textit {\_Z}}+108 x^{2}-27 \ln \left (\frac {2 \,{\mathrm e}^{\textit {\_Z}}-9}{\left (x +1\right )^{4}}\right )-81 c_1 +27 \textit {\_Z} -108 x +27\right )}-18 x^{2}-9}{4 \,{\mathrm e}^{\operatorname {RootOf}\left (8 x^{3} {\mathrm e}^{\textit {\_Z}}-24 \,{\mathrm e}^{\textit {\_Z}} x^{2}-36 x^{3}+6 \ln \left (\frac {2 \,{\mathrm e}^{\textit {\_Z}}-9}{\left (x +1\right )^{4}}\right ) {\mathrm e}^{\textit {\_Z}}+18 c_1 \,{\mathrm e}^{\textit {\_Z}}-6 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+24 x \,{\mathrm e}^{\textit {\_Z}}+108 x^{2}-27 \ln \left (\frac {2 \,{\mathrm e}^{\textit {\_Z}}-9}{\left (x +1\right )^{4}}\right )-81 c_1 +27 \textit {\_Z} -108 x +27\right )}-18}
\]
✓ Mathematica. Time used: 1.101 (sec). Leaf size: 140
ode=D[y[x],x] == (x*(-1 - x + x^2 + 2*x^4 - 2*x^2*y[x]))/((1 + x)*(x^2 - y[x]));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x\frac {4\ 2^{2/3} \left (K[2]^3 \left (K[2]^2-K[2]-1\right )^3\right )^{2/3}}{9 K[2] \left (K[2]^3-2 K[2]-1\right )}dK[2]+c_1=\int _1^{\frac {x \left (x^2-x-1\right ) \left (2 x^2-2 y(x)+3\right )}{2 \sqrt [3]{2} \sqrt [3]{x^3 \left (x^2-x-1\right )^3} \left (x^2-y(x)\right )}}\frac {1}{K[1]^3-\frac {3 K[1]}{2^{2/3}}+1}dK[1],y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x*(2*x**4 - 2*x**2*y(x) + x**2 - x - 1)/((x + 1)*(x**2 - y(x))) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out