54.2.137 problem 714

Internal problem ID [12011]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 714
Date solved : Tuesday, September 30, 2025 at 11:54:45 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }&=-\frac {y \left (-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}+y x^{2} \ln \left (x \right )+x^{3} y-x \ln \left (x \right )-x^{2}\right )}{\left (-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}\right ) x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 102

ode:=diff(y(x),x) = -y(x)*(-ln(1/x)+exp(x)+y(x)*x^2*ln(x)+x^3*y(x)-x*ln(x)-x^2)/(-ln(1/x)+exp(x))/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\int \frac {x \ln \left (x \right )+x^{2}+\ln \left (\frac {1}{x}\right )-{\mathrm e}^{x}}{\left (\ln \left (\frac {1}{x}\right )-{\mathrm e}^{x}\right ) x}d x}}{-\int \frac {{\mathrm e}^{-\int \frac {x \ln \left (x \right )+x^{2}+\ln \left (\frac {1}{x}\right )-{\mathrm e}^{x}}{\left (\ln \left (\frac {1}{x}\right )-{\mathrm e}^{x}\right ) x}d x} x \left (\ln \left (x \right )+x \right )}{\ln \left (\frac {1}{x}\right )-{\mathrm e}^{x}}d x +c_1} \]

✓ Mathematica. Time used: 0.747 (sec). Leaf size: 150

ode=D[y[x],x] == -((y[x]*(E^x - x^2 - Log[x^(-1)] - x*Log[x] + x^3*y[x] + x^2*Log[x]*y[x]))/(x*(E^x - Log[x^(-1)]))); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\exp \left (\int _1^x\frac {K[1]^2+\log (K[1]) K[1]-e^{K[1]}+\log \left (\frac {1}{K[1]}\right )}{K[1] \left (e^{K[1]}-\log \left (\frac {1}{K[1]}\right )\right )}dK[1]\right )}{-\int _1^x-\frac {\exp \left (\int _1^{K[2]}\frac {K[1]^2+\log (K[1]) K[1]-e^{K[1]}+\log \left (\frac {1}{K[1]}\right )}{K[1] \left (e^{K[1]}-\log \left (\frac {1}{K[1]}\right )\right )}dK[1]\right ) K[2] (K[2]+\log (K[2]))}{e^{K[2]}-\log \left (\frac {1}{K[2]}\right )}dK[2]+c_1}\\ y(x)&\to 0 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) + (x**3*y(x) + x**2*y(x)*log(x) - x**2 - x*log(x) + exp(x) - log(1/x))*y(x)/(x*(exp(x) - log(1/x))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Invalid NaN comparison