82.3.34 problem 25-34

Internal problem ID [21817]
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 : 25-34
Date solved : Thursday, October 02, 2025 at 08:02:33 PM
CAS classification : [[_high_order, _exact, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.098 (sec). Leaf size: 96

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

\[ y(x)\to \frac {1}{4} (\log (x)+1) \left (-2 \log ^2(x)-5 \log (x)-5\right )+\frac {1}{4} (\log (x)+1) \left (\log ^2(x)+1\right )+\frac {1}{8} \left (2 \log ^3(x)+12 \log ^2(x)+25 \log (x)+25\right )+\frac {1}{8} (\log (x)-1)+\frac {c_1}{x}+c_4 x+c_3 x \left (\log ^2(x)+1\right )+c_2 x (\log (x)+1) \]

✗ 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