20.25.2 problem 3

Internal problem ID [4007]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 3
Date solved : Sunday, March 30, 2025 at 02:14:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 35

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

\[ y = c_1 \,x^{-i} \left (1+\left (-\frac {1}{4}+\frac {i}{4}\right ) x^{2}+\left (-\frac {1}{80}+\frac {7 i}{80}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{i} \left (1+\left (-\frac {1}{4}-\frac {i}{4}\right ) x^{2}+\left (-\frac {1}{80}-\frac {7 i}{80}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 70

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

\[ y(x)\to \left (\frac {1}{80}+\frac {3 i}{80}\right ) c_2 x^{-i} \left ((2+i) x^4+(4+8 i) x^2+(8-24 i)\right )-\left (\frac {3}{80}+\frac {i}{80}\right ) c_1 x^i \left ((1+2 i) x^4+(8+4 i) x^2-(24-8 i)\right ) \]

✗ Sympy

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

ValueError : Expected Expr or iterable but got None