60.3.395 problem 1412

Internal problem ID [11391]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1412
Date solved : Sunday, March 30, 2025 at 08:19:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

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

\[ y = \sinh \left (x \left (-1+\ln \left (x \right )\right )\right ) c_1 +\cosh \left (x \left (-1+\ln \left (x \right )\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 29

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

\[ y(x)\to c_1 \cosh (x (\log (x)-1))+i c_2 \sinh (x (\log (x)-1)) \]

✗ Sympy

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

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