6.9.20 problem 20

Internal problem ID [1776]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 05:19:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\ln \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 12

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

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 15

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

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

✗ Sympy

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

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