60.2.339 problem 917

Internal problem ID [10913]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 917
Date solved : Friday, March 14, 2025 at 02:48:21 AM
CAS classification : [NONE]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 38

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

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

✓ Mathematica. Time used: 0.492 (sec). Leaf size: 35

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

\begin{align*} y(x)\to \frac {e^{-\frac {x}{x-\log (x+1)-c_1}}}{x} \\ y(x)\to \frac {1}{x} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x*log(x)**2 + 2*x*log(x)*log(y(x)) + x*log(x) + x*log(y(x))**2 + x*log(y(x)) - x + log(x) + log(y(x)) - 1)*y(x)/(x*(x + 1)) cannot be solved by the lie group method