9.19 problem 10, using series method

Internal problem ID [4402]

Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section: Chapter 12, Series Solutions of Differential Equations. Section 1. Miscellaneous problems. page 564
Problem number: 10, using series method.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]

Solve \begin {gather*} \boxed {y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.002 (sec). Leaf size: 30

Order:=6; 
dsolve(diff(y(x),x$2)-4*x*diff(y(x),x)+(4*x^2-2)*y(x)=0,y(x),type='series',x=0);
 

\[ y \relax (x ) = \left (1+x^{2}+\frac {1}{2} x^{4}\right ) y \relax (0)+\left (x +x^{3}+\frac {1}{2} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]

Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 34

AsymptoticDSolveValue[y''[x]-4*x*y'[x]+(4*x^2-2)*y[x]==0,y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{2}+x^3+x\right )+c_1 \left (\frac {x^4}{2}+x^2+1\right ) \]