47.5.8 problem 8

Internal problem ID [7497]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.3.4 problems. page 104
Problem number : 8
Date solved : Sunday, March 30, 2025 at 12:10:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 22

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

\[ y = \frac {\ln \left (x \right )^{2} x}{2}-\left (x +c_1 \right ) \ln \left (x \right )+c_2 x \]

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 27

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

\[ y(x)\to \frac {1}{2} x \log ^2(x)+c_1 x-(x+c_2) \log (x) \]

✗ Sympy

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

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