Internal problem ID [11926]
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: 25.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\left (x^{2}+1\right ) y=0} \] 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*diff(y(x),x)+(x^2+1)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 65
AsymptoticDSolveValue[x^2*y''[x]-x*y'[x]+(x^2+1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (x \left (\frac {x^2}{4}-\frac {3 x^4}{128}\right )+x \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]