82.1.8 problem 23-11 (b)

Internal problem ID [21750]
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 (b)
Date solved : Thursday, October 02, 2025 at 08:01:48 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*} 1 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 46

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=1);
 

\[ y = \left (c_2 \ln \left (x -1\right )+c_1 \right ) \left (1-\left (x -1\right )+\frac {3}{4} \left (x -1\right )^{2}-\frac {17}{36} \left (x -1\right )^{3}+\frac {167}{576} \left (x -1\right )^{4}-\frac {299}{1600} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+\left (\left (x -1\right )-\left (x -1\right )^{2}+\frac {73}{108} \left (x -1\right )^{3}-\frac {1415}{3456} \left (x -1\right )^{4}+\frac {36299}{144000} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 141

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,1,5}]
 

\[ y(x)\to c_1 \left (-\frac {299 (x-1)^5}{1600}+\frac {167}{576} (x-1)^4-\frac {17}{36} (x-1)^3+\frac {3}{4} (x-1)^2-x+2\right )+c_2 \left (\frac {36299 (x-1)^5}{144000}-\frac {1415 (x-1)^4}{3456}+\frac {73}{108} (x-1)^3-(x-1)^2+2 (x-1)-x+\left (-\frac {299 (x-1)^5}{1600}+\frac {167}{576} (x-1)^4-\frac {17}{36} (x-1)^3+\frac {3}{4} (x-1)^2-x+2\right ) \log (x-1)+1\right ) \]

✓ Sympy. Time used: 0.321 (sec). Leaf size: 7

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_regular",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{1} + O\left (x^{6}\right ) \]