69.20.34 problem 673

Internal problem ID [18455]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 673
Date solved : Thursday, October 02, 2025 at 03:12:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} \left (\ln \left (x \right )-1\right ) y^{\prime \prime }-x^{2} y^{\prime }+y x&=2 \ln \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (\infty \right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 16

ode:=x^3*(ln(x)-1)*diff(diff(y(x),x),x)-x^2*diff(y(x),x)+x*y(x) = 2*ln(x); 
ic:=[y(infinity) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {-c_1 \ln \left (x \right ) x +1}{x} \]

✗ Mathematica

ode=x^3*(Log[x]-1)*D[y[x],{x,2}]-x^2*D[y[x],x]+x*y[x]==2*Log[x]; 
ic={y[Infinity]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

{}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*(log(x) - 1)*Derivative(y(x), (x, 2)) - x**2*Derivative(y(x), x) + x*y(x) - 2*log(x),0) 
ics = {y(oo): 0} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**3*(log(x) - 1)*Derivative(y(x), (x, 2)) + x*y(x) - 2*log(x))/x**2 cannot be solved by the factorable group method