43.1.6 problem 7.2.6

Internal problem ID [6851]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.2.1 Exercises. page 290
Problem number : 7.2.6
Date solved : Monday, January 27, 2025 at 02:31:41 PM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 65

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

\[ y = \left (1-\frac {p^{2} x^{2}}{2}+\frac {p^{2} \left (p^{2}-4\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (p^{2}-1\right ) x^{3}}{6}+\frac {\left (p^{4}-10 p^{2}+9\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 88

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

\[ y(x)\to c_2 \left (\frac {p^4 x^5}{120}-\frac {p^2 x^5}{12}-\frac {p^2 x^3}{6}+\frac {3 x^5}{40}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {p^4 x^4}{24}-\frac {p^2 x^4}{6}-\frac {p^2 x^2}{2}+1\right ) \]