82.2.8 problem 24-8

Internal problem ID [21769]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 24. Power series about an ordinary point. Page 719
Problem number : 24-8
Date solved : Thursday, October 02, 2025 at 08:01:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (x -1\right )^{2} y^{\prime }-4 \left (x -1\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 34

Order:=6; 
ode:=diff(diff(y(x),x),x)+(x-1)^2*diff(y(x),x)-4*(x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1+\frac {2 \left (x -1\right )^{3}}{3}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{4}}{4}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 33

ode=D[y[x],{x,2}]+(x-1)^2*D[y[x],x]-4*(x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_1 \left (\frac {2}{3} (x-1)^3+1\right )+c_2 \left (\frac {1}{4} (x-1)^4+x-1\right ) \]

✓ Sympy. Time used: 0.238 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 1)**2*Derivative(y(x), x) - (4*x - 4)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

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