87.14.15 problem 15

Internal problem ID [23534]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 15
Date solved : Thursday, October 02, 2025 at 09:42:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (1-2 x \ln \left (x \right )\right ) y^{\prime \prime }+\left (1+4 x^{2} \ln \left (x \right )\right ) y^{\prime }-\left (4 x +2\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: 0.036 (sec). Leaf size: 15

ode:=x*(1-2*x*ln(x))*diff(diff(y(x),x),x)+(1+4*x^2*ln(x))*diff(y(x),x)-(2+4*x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \ln \left (x \right )+c_2 \,{\mathrm e}^{2 x} \]

✗ Mathematica

ode=x*(1-2*x*Log[x])*D[y[x],{x,2}]+(1+4*x^2*Log[x])*D[y[x],x]-(2+4*x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(-2*x*log(x) + 1)*Derivative(y(x), (x, 2)) - (4*x + 2)*y(x) + (4*x**2*log(x) + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (2*x**2*log(x)*Derivative(y(x), (x, 2)) +