60.2.125 problem 701

Internal problem ID [10699]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 701
Date solved : Friday, March 14, 2025 at 02:21:35 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=\frac {2 x \,{\mathrm e}^{x}-2 x -\ln \left (x \right )-1+x^{4} \ln \left (x \right )+x^{4}-2 y x^{2} \ln \left (x \right )-2 x^{2} y+y^{2} \ln \left (x \right )+y^{2}}{{\mathrm e}^{x}-1} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 71

ode:=diff(y(x),x) = (2*x*exp(x)-2*x-ln(x)-1+x^4*ln(x)+x^4-2*y(x)*x^2*ln(x)-2*x^2*y(x)+y(x)^2*ln(x)+y(x)^2)/(exp(x)-1); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-x^{2} {\mathrm e}^{2 \left (\int \frac {1+\ln \left (x \right )}{-1+{\mathrm e}^{x}}d x \right )}+c_{1} x^{2}+{\mathrm e}^{2 \left (\int \frac {1+\ln \left (x \right )}{-1+{\mathrm e}^{x}}d x \right )}+c_{1}}{-{\mathrm e}^{2 \left (\int \frac {1+\ln \left (x \right )}{-1+{\mathrm e}^{x}}d x \right )}+c_{1}} \]

✓ Mathematica. Time used: 1.487 (sec). Leaf size: 97

ode=D[y[x],x] == (-1 - 2*x + 2*E^x*x + x^4 - Log[x] + x^4*Log[x] - 2*x^2*y[x] - 2*x^2*Log[x]*y[x] + y[x]^2 + Log[x]*y[x]^2)/(-1 + E^x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\frac {2 (\log (K[5])+1)}{-1+e^{K[5]}}dK[5]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[6]}\frac {2 (\log (K[5])+1)}{-1+e^{K[5]}}dK[5]\right ) (\log (K[6])+1)}{-1+e^{K[6]}}dK[6]+c_1}+x^2+1 \\ y(x)\to x^2+1 \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**4*log(x) + x**4 - 2*x**2*y(x)*log(x) - 2*x**2*y(x) + 2*x*exp(x) - 2*x + y(x)**2*log(x) + y(x)**2 - log(x) - 1)/(exp(x) - 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

IndexError : Index out of range: a[1]