73.23.26 problem 33.10

Internal problem ID [15672]
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 : 33.10
Date solved : Tuesday, January 28, 2025 at 08:03:58 AM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\lambda y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 53

Order:=6; 
dsolve((1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+lambda*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {\lambda \,x^{2}}{2}+\frac {\lambda \left (\lambda -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\lambda -2\right ) x^{3}}{6}+\frac {\left (\lambda -2\right ) \left (-12+\lambda \right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 80

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

\[ y(x)\to c_2 \left (\frac {\lambda ^2 x^5}{120}-\frac {7 \lambda x^5}{60}+\frac {x^5}{5}-\frac {\lambda x^3}{6}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {\lambda ^2 x^4}{24}-\frac {\lambda x^4}{4}-\frac {\lambda x^2}{2}+1\right ) \]