82.3.26 problem 25-26

Internal problem ID [21809]
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 : 25-26
Date solved : Thursday, October 02, 2025 at 08:02:28 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

\begin{align*} \infty \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 114

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+(3*x-1)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=infinity);
 

\[ y = \frac {7200 \left (O\left (\frac {1}{x^{6}}\right ) x^{5}+x^{5}-x^{4}+\frac {x^{3}}{2}-\frac {x^{2}}{6}+\frac {x}{24}-\frac {1}{120}\right ) c_2 \ln \left (\frac {1}{x}\right )+7200 x^{5} \left (c_1 +c_2 \right ) O\left (\frac {1}{x^{6}}\right )+7200 c_1 \,x^{5}+\left (-7200 c_1 +7200 c_2 \right ) x^{4}+\left (3600 c_1 -5400 c_2 \right ) x^{3}+\left (-1200 c_1 +2200 c_2 \right ) x^{2}+\left (300 c_1 -625 c_2 \right ) x -60 c_1 +137 c_2}{7200 x^{6}} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 128

ode=x^2*D[y[x],{x,2}]+(3*x-1)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,Infinity,5}]
 

\[ y(x)\to c_1 \left (-\frac {1}{120 x^6}+\frac {1}{24 x^5}-\frac {1}{6 x^4}+\frac {1}{2 x^3}-\frac {1}{x^2}+\frac {1}{x}\right )+c_2 \left (\frac {137}{7200 x^6}+\frac {\log (x)}{120 x^6}-\frac {25}{288 x^5}-\frac {\log (x)}{24 x^5}+\frac {11}{36 x^4}+\frac {\log (x)}{6 x^4}-\frac {3}{4 x^3}-\frac {\log (x)}{2 x^3}+\frac {1}{x^2}+\frac {\log (x)}{x^2}-\frac {\log (x)}{x}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (3*x - 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=oo,n=6)
 

ValueError : ODE x**2*Derivative(y(x), (x, 2)) + (3*x - 1)*Derivative(y(x), x) + y(x) does not match hint 2nd_power_series_regular