73.23.18 problem 33.5 (f)

Internal problem ID [15664]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.5 (f)
Date solved : Tuesday, January 28, 2025 at 08:03:51 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-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((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (-2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}-\frac {1}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 33

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

\[ y(x)\to c_1 \left (1-2 x^2\right )+c_2 \left (-\frac {x^5}{8}-\frac {x^3}{2}+x\right ) \]