30.14.11 problem 11

Internal problem ID [7660]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 04:55:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 76

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

\[ y = \left (1-\frac {\left (x -2\right )^{2}}{8}+\frac {\left (x -2\right )^{3}}{32}-\frac {3 \left (x -2\right )^{4}}{512}+\frac {\left (x -2\right )^{5}}{2048}\right ) y \left (2\right )+\left (x -2+\frac {\left (x -2\right )^{2}}{8}-\frac {7 \left (x -2\right )^{3}}{96}+\frac {37 \left (x -2\right )^{4}}{1536}-\frac {211 \left (x -2\right )^{5}}{30720}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (\frac {(x-2)^5}{2048}-\frac {3}{512} (x-2)^4+\frac {1}{32} (x-2)^3-\frac {1}{8} (x-2)^2+1\right )+c_2 \left (-\frac {211 (x-2)^5}{30720}+\frac {37 (x-2)^4}{1536}-\frac {7}{96} (x-2)^3+\frac {1}{8} (x-2)^2+x-2\right ) \]

✓ Sympy. Time used: 0.254 (sec). Leaf size: 60

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

\[ y{\left (x \right )} = C_{2} \left (x + \frac {37 \left (x - 2\right )^{4}}{1536} - \frac {7 \left (x - 2\right )^{3}}{96} + \frac {\left (x - 2\right )^{2}}{8} - 2\right ) + C_{1} \left (- \frac {3 \left (x - 2\right )^{4}}{512} + \frac {\left (x - 2\right )^{3}}{32} - \frac {\left (x - 2\right )^{2}}{8} + 1\right ) + O\left (x^{6}\right ) \]