82.1.7 problem 23-11 (a)

Internal problem ID [21749]
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 (a)
Date solved : Thursday, October 02, 2025 at 08:01:46 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*} 0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 58

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=0);
 

\[ y = c_1 x \left (1+\frac {1}{2} x +\frac {5}{12} x^{2}+\frac {47}{144} x^{3}+\frac {791}{2880} x^{4}+\frac {20371}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (x +\frac {1}{2} x^{2}+\frac {5}{12} x^{3}+\frac {47}{144} x^{4}+\frac {791}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {1}{4} x^{2}-\frac {1}{18} x^{3}-\frac {85}{1728} x^{4}-\frac {2491}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 87

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,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{144} x \left (47 x^3+60 x^2+72 x+144\right ) \log (x)+\frac {-1213 x^4-1536 x^3-2160 x^2-3456 x+1728}{1728}\right )+c_2 \left (\frac {791 x^5}{2880}+\frac {47 x^4}{144}+\frac {5 x^3}{12}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.325 (sec). Leaf size: 10

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=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x + C_{1} + O\left (x^{6}\right ) \]