Internal problem ID [11909]
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: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\left (2 x^{2}+\frac {5}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 35
Order:=6; dsolve(x^2*diff(y(x),x$2)-x*diff(y(x),x)+(2*x^2+5/9)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {3}{2} x^{2}+\frac {9}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {5}{3}} \left (1-\frac {3}{10} x^{2}+\frac {9}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 52
AsymptoticDSolveValue[x^2*y''[x]-x*y'[x]+(2*x^2+5/9)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {9 x^4}{32}-\frac {3 x^2}{2}+1\right ) \sqrt [3]{x}+c_1 \left (\frac {9 x^4}{320}-\frac {3 x^2}{10}+1\right ) x^{5/3} \]