82.3.33 problem 25-33

Internal problem ID [21816]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 25. Power series about a singular point. Page 762
Problem number : 25-33
Date solved : Thursday, October 02, 2025 at 08:02:33 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime }-4 x^{2} y^{\prime \prime }+8 x y^{\prime }-8 y&=4 \ln \left (x \right ) \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✗ Maple

Order:=6; 
ode:=x^3*diff(diff(diff(y(x),x),x),x)-4*x^2*diff(diff(y(x),x),x)+8*x*diff(y(x),x)-8*y(x) = 4*ln(x); 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 48

ode=x^3*D[y[x],{x,3}]-4*x^2*D[y[x],{x,2}]+8*x*D[y[x],x]-8*y[x]==4*Log[x]; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_3 x^4+c_2 x^2+\frac {1}{24} (-4 \log (x)-1)-\frac {4}{3} (\log (x)+1)+\frac {1}{2} (2 \log (x)+1)+c_1 x \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3 + x**2*Derivative(y(x), (x, 2)) - 2*x*Derivative(y(x), x) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

Series solution not supported for ode of order > 2