82.1.9 problem 23-11 (c)

Internal problem ID [21751]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 23. Power series. Page 695
Problem number : 23-11 (c)
Date solved : Thursday, October 02, 2025 at 08:01:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 76

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

\[ y = \left (1-\frac {\left (-2+x \right )^{2}}{4}+\frac {7 \left (-2+x \right )^{3}}{24}-\frac {\left (-2+x \right )^{4}}{4}+\frac {191 \left (-2+x \right )^{5}}{960}\right ) y \left (2\right )+\left (-2+x -\left (-2+x \right )^{2}+\frac {3 \left (-2+x \right )^{3}}{4}-\frac {25 \left (-2+x \right )^{4}}{48}+\frac {89 \left (-2+x \right )^{5}}{240}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {191}{960} (x-2)^5-\frac {1}{4} (x-2)^4+\frac {7}{24} (x-2)^3-\frac {1}{4} (x-2)^2+1\right )+c_2 \left (\frac {89}{240} (x-2)^5-\frac {25}{48} (x-2)^4+\frac {3}{4} (x-2)^3-(x-2)^2+x-2\right ) \]

✓ Sympy. Time used: 0.322 (sec). Leaf size: 58

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

\[ y{\left (x \right )} = C_{2} \left (x - \frac {25 \left (x - 2\right )^{4}}{48} + \frac {3 \left (x - 2\right )^{3}}{4} - \left (x - 2\right )^{2} - 2\right ) + C_{1} \left (- \frac {\left (x - 2\right )^{4}}{4} + \frac {7 \left (x - 2\right )^{3}}{24} - \frac {\left (x - 2\right )^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]