30.13.6 problem 6

Internal problem ID [7638]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 04:55:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 59

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

\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}-\frac {1}{15} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{60} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 70

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

\[ y(x)\to c_1 \left (-\frac {x^5}{15}+\frac {x^4}{12}-\frac {x^3}{6}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^5}{60}-\frac {x^4}{12}+\frac {x^3}{6}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.344 (sec). Leaf size: 54

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

\[ y{\left (x \right )} = \frac {x^{4} r{\left (3 \right )}}{4} + \frac {x^{5} r{\left (3 \right )}}{4} + C_{2} \left (\frac {x^{5}}{60} + \frac {x^{4}}{12} + 1\right ) + C_{1} x \left (- \frac {x^{4}}{40} - \frac {x^{3}}{8} + \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]