54.2.276 problem 854

Internal problem ID [12150]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 854
Date solved : Sunday, October 12, 2025 at 02:24:01 AM
CAS classification : [NONE]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 21

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

\[ y = \frac {{\mathrm e}^{-\frac {3 x}{x^{3}+3 c_1}}}{x} \]

✓ Mathematica. Time used: 0.212 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to \frac {e^{-\frac {3 x}{x^3+3 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**2*log(x)**2 + 2*x**2*log(x)*log(y(x)) + x**2*log(y(x))**2 + log(x) + log(y(x)) - 1)*y(x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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