54.3.270 problem 1286

Internal problem ID [12565]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1286
Date solved : Wednesday, October 01, 2025 at 02:06:46 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} \left (3 x -1\right )^{2} y^{\prime \prime }+3 \left (3 x -1\right ) y^{\prime }-9 y-\ln \left (3 x -1\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

ode:=(3*x-1)^2*diff(diff(y(x),x),x)+3*(3*x-1)*diff(y(x),x)-9*y(x)-ln(3*x-1)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1}{3 x -1}+\left (3 x -1\right ) c_2 -\frac {\ln \left (3 x -1\right )^{2}}{9}-\frac {2}{9} \]

✓ Mathematica. Time used: 0.131 (sec). Leaf size: 145

ode=-Log[-1 + 3*x]^2 - 9*y[x] + 3*(-1 + 3*x)*D[y[x],x] + (-1 + 3*x)^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\left (-9 x^2+6 x-2\right ) \int _1^x\frac {K[1] (3 K[1]-2) \log ^2(3 K[1]-1)}{2 (1-3 K[1])^2}dK[1]-3 i (3 x-2) x \int _1^x\frac {i \left (9 K[2]^2-6 K[2]+2\right ) \log ^2(3 K[2]-1)}{6 (1-3 K[2])^2}dK[2]-9 c_1 x^2-9 i c_2 x^2+6 c_1 x+6 i c_2 x-2 c_1}{6 x-2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x - 1)**2*Derivative(y(x), (x, 2)) + (9*x - 3)*Derivative(y(x), x) - 9*y(x) - log(3*x - 1)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-9*x**2*Derivative(y(x), (x, 2)) + 6*x*Derivative(y(x), (x, 2)) + 9*y(x) + log(3*x - 1)**2 - Derivative(y(x), (x, 2)))/(3*(3*x - 1)) cannot be solved by the factorable group method