Internal problem ID [2409]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\frac {x y^{\prime }}{\left (1-x^{2}\right )^{2}}+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 45
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);
\[ y \relax (x ) = c_{1} x^{-i} \left (1+\left (-\frac {1}{4}+\frac {i}{4}\right ) x^{2}+\left (-\frac {1}{80}+\frac {7 i}{80}\right ) x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{i} \left (1+\left (-\frac {1}{4}-\frac {i}{4}\right ) x^{2}+\left (-\frac {1}{80}-\frac {7 i}{80}\right ) x^{4}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 70
AsymptoticDSolveValue[x^2*y''[x]+x/(1-x^2)^2*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \left (\frac {1}{80}+\frac {3 i}{80}\right ) c_2 x^{-i} \left ((2+i) x^4+(4+8 i) x^2+(8-24 i)\right )-\left (\frac {3}{80}+\frac {i}{80}\right ) c_1 x^i \left ((1+2 i) x^4+(8+4 i) x^2-(24-8 i)\right ) \]