Internal problem ID [11915]
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: 14.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{4}+x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 29
Order:=6; dsolve(x^2*diff(y(x),x$2)+(x^4+x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{15} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-2-\frac {2}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 30
AsymptoticDSolveValue[x^2*y''[x]+(x^4+x)*y'[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (x-\frac {x^4}{15}\right )+c_1 \left (\frac {x^2}{3}+\frac {1}{x}\right ) \]