54.3.113 problem 1127

Internal problem ID [12408]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1127
Date solved : Wednesday, October 01, 2025 at 01:44:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (2 a x \ln \left (x \right )+1\right ) y^{\prime }+\left (a^{2} x \ln \left (x \right )^{2}+a \ln \left (x \right )+a \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 25

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

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

✗ Sympy

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

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