30.14.9 problem 9

Internal problem ID [7658]
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 : 9
Date solved : Tuesday, September 30, 2025 at 04:55:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 46

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

\[ y = \left (1+\left (x -1\right )^{2}+\frac {\left (x -1\right )^{4}}{3}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{3}}{3}+\frac {2 \left (x -1\right )^{5}}{15}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 47

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

\[ y(x)\to c_1 \left (\frac {1}{3} (x-1)^4+(x-1)^2+1\right )+c_2 \left (\frac {2}{15} (x-1)^5+\frac {1}{3} (x-1)^3+x-1\right ) \]

✓ Sympy. Time used: 0.234 (sec). Leaf size: 32

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

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