Internal problem ID [11927]
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: 26.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\left (x^{2}-3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 47
Order:=6; dsolve(x^2*diff(y(x),x$2)-x*diff(y(x),x)+(x^2-3)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 52
AsymptoticDSolveValue[x^2*y''[x]-x*y'[x]+(x^2-3)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {\left (x^2+8\right )^2}{64 x}-\frac {1}{16} x^3 \log (x)\right )+c_2 \left (\frac {x^7}{384}-\frac {x^5}{12}+x^3\right ) \]