54.3.396 problem 1413

Internal problem ID [12691]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1413
Date solved : Wednesday, October 01, 2025 at 02:20:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 12

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

\[ y = c_1 x +c_2 \ln \left (x \right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 16

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

\begin{align*} y(x)&\to c_1 x-c_2 \log (x) \end{align*}

✗ Sympy

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

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