Internal problem ID [11922]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius).
Exercises page 251
Problem number: 21.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +8 y \left (x^{2}-1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 35
Order:=6; dsolve(x^2*diff(y(x),x$2)-x*diff(y(x),x)+8*(x^2-1)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{4} \left (1-\frac {1}{2} x^{2}+\frac {1}{10} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-86400-86400 x^{2}-86400 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 36
AsymptoticDSolveValue[x^2*y''[x]-x*y'[x]+8*(x^2-1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (x^2+\frac {1}{x^2}+1\right )+c_2 \left (\frac {x^8}{10}-\frac {x^6}{2}+x^4\right ) \]