65.13.6 problem 20.2 (iv) (k=2)

Internal problem ID [13812]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 20, Series solutions of second order linear equations. Exercises page 195
Problem number : 20.2 (iv) (k=2)
Date solved : Tuesday, January 28, 2025 at 06:04:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 34

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

\[ y = \left (-2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}-\frac {1}{30} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 33

AsymptoticDSolveValue[D[y[x],{x,2}]-2*x*D[y[x],x]+4*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (1-2 x^2\right )+c_2 \left (-\frac {x^5}{30}-\frac {x^3}{3}+x\right ) \]