60.2.130 problem 706
Internal
problem
ID
[10704]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
706
Date
solved
:
Sunday, March 30, 2025 at 06:23:05 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} y^{\prime }&=-\frac {\left (-\ln \left (-1+y\right )+\ln \left (y+1\right )+2 \ln \left (x \right )\right ) x \left (y+1\right )^{2}}{8} \end{align*}
✓ Maple. Time used: 0.523 (sec). Leaf size: 120
ode:=diff(y(x),x) = -1/8*(-ln(-1+y(x))+ln(1+y(x))+2*ln(x))*x*(1+y(x))^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} \ln \left (x^{2} \left ({\mathrm e}^{\textit {\_Z}}+2\right )\right ) x^{2}+\textit {\_Z} \,x^{2} {\mathrm e}^{\textit {\_Z}}-2 \ln \left (x^{2} \left ({\mathrm e}^{\textit {\_Z}}+2\right )\right ) x^{2}+2 \textit {\_Z} \,x^{2}-8 \,{\mathrm e}^{\textit {\_Z}}\right )}+1 \\
\int _{\textit {\_b}}^{y}-\frac {1}{2 \left (-\frac {x^{2} \left (\textit {\_a} +1\right ) \ln \left (-1+\textit {\_a} \right )}{2}+\frac {x^{2} \left (\textit {\_a} +1\right ) \ln \left (\textit {\_a} +1\right )}{2}+x^{2} \left (\textit {\_a} +1\right ) \ln \left (x \right )+4 \textit {\_a} -4\right ) \left (\textit {\_a} +1\right )}d \textit {\_a} -\frac {\ln \left (x \right )}{8}-c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 1.73 (sec). Leaf size: 610
ode=D[y[x],x] == -1/8*(x*(2*Log[x] - Log[-1 + y[x]] + Log[1 + y[x]])*(1 + y[x])^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (\frac {-2 \log (x) x^2+\log (K[2]-1) x^2-\log (K[2]+1) x^2-8}{2 \left (2 \log (x) x^2-\log (K[2]-1) x^2+\log (K[2]+1) x^2+K[2] \left (2 \log (x) x^2-\log (K[2]-1) x^2+\log (K[2]+1) x^2+8\right )-8\right )}-\int _1^x\left (-\frac {K[1] (K[2]+1) \left (\frac {1}{K[2]+1}-\frac {1}{K[2]-1}\right )}{2 K[2] \log (K[1]) K[1]^2+2 \log (K[1]) K[1]^2-K[2] \log (K[2]-1) K[1]^2-\log (K[2]-1) K[1]^2+K[2] \log (K[2]+1) K[1]^2+\log (K[2]+1) K[1]^2+8 K[2]-8}-\frac {K[1] (2 \log (K[1])-\log (K[2]-1)+\log (K[2]+1))}{2 K[2] \log (K[1]) K[1]^2+2 \log (K[1]) K[1]^2-K[2] \log (K[2]-1) K[1]^2-\log (K[2]-1) K[1]^2+K[2] \log (K[2]+1) K[1]^2+\log (K[2]+1) K[1]^2+8 K[2]-8}+\frac {K[1] (K[2]+1) (2 \log (K[1])-\log (K[2]-1)+\log (K[2]+1)) \left (-\frac {K[2] K[1]^2}{K[2]-1}+2 \log (K[1]) K[1]^2-\log (K[2]-1) K[1]^2+\log (K[2]+1) K[1]^2-\frac {K[1]^2}{K[2]-1}+\frac {K[2] K[1]^2}{K[2]+1}+\frac {K[1]^2}{K[2]+1}+8\right )}{\left (2 K[2] \log (K[1]) K[1]^2+2 \log (K[1]) K[1]^2-K[2] \log (K[2]-1) K[1]^2-\log (K[2]-1) K[1]^2+K[2] \log (K[2]+1) K[1]^2+\log (K[2]+1) K[1]^2+8 K[2]-8\right )^2}\right )dK[1]+\frac {1}{2 (K[2]+1)}\right )dK[2]+\int _1^x-\frac {K[1] (2 \log (K[1])-\log (y(x)-1)+\log (y(x)+1)) (y(x)+1)}{2 \log (K[1]) K[1]^2-\log (y(x)-1) K[1]^2+\log (y(x)+1) K[1]^2+2 \log (K[1]) y(x) K[1]^2-\log (y(x)-1) y(x) K[1]^2+\log (y(x)+1) y(x) K[1]^2+8 y(x)-8}dK[1]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(y(x) + 1)**2*(2*log(x) - log(y(x) - 1) + log(y(x) + 1))/8 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -x*(-2*y(x)**2*log(x) + y(x)**2*log(y(x) - 1) - y(x)**2*log(y(x) + 1) - 4*y(x)*log(x) + 2*y(x)*log(y(x) - 1) - 2*y(x)*log(y(x) + 1) - 2*log(x) + log(y(x) - 1) - log(y(x) + 1))/8 + Derivative(y(x), x) cannot be solved by the factorable group method