60.1.191 problem 194

Internal problem ID [10205]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 194
Date solved : Sunday, March 30, 2025 at 03:30:33 PM
CAS classification : [_Riccati]

\begin{align*} x y^{\prime } \ln \left (x \right )-y^{2} \ln \left (x \right )-\left (2 \ln \left (x \right )^{2}+1\right ) y-\ln \left (x \right )^{3}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 23

ode:=x*diff(y(x),x)*ln(x)-y(x)^2*ln(x)-(2*ln(x)^2+1)*y(x)-ln(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\ln \left (x \right ) \left (\ln \left (x \right )^{2}+c_1 +2\right )}{\ln \left (x \right )^{2}+c_1} \]

✓ Mathematica. Time used: 0.305 (sec). Leaf size: 38

ode=x*D[y[x],x]*Log[x] - y[x]^2*Log[x] - (2*Log[x]^2+1)*y[x] - Log[x]^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\log (x) \left (\log ^2(x)+2+2 c_1\right )}{\log ^2(x)+2 c_1} \\ y(x)\to -\log (x) \\ \end{align*}

✗ Sympy

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

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