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 : Monday, January 27, 2025 at 08:06:05 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*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x/(1-x^2)^2*D[y[x],x]+y[x]==0,y[x],{x,0,"6"-1}]
 

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