Internal problem ID [4755]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+p^{2} y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 71
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);
\[ y \relax (x ) = \left (1-\frac {p^{2} x^{2}}{2}+\frac {p^{2} \left (p^{2}-4\right ) x^{4}}{24}\right ) y \relax (0)+\left (x +\frac {\left (-p^{2}+1\right ) x^{3}}{6}+\frac {\left (p^{4}-10 p^{2}+9\right ) x^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 88
AsymptoticDSolveValue[(1-x^2)*y''[x]-x*y'[x]+p^2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {p^4 x^5}{120}-\frac {p^2 x^5}{12}-\frac {p^2 x^3}{6}+\frac {3 x^5}{40}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {p^4 x^4}{24}-\frac {p^2 x^4}{6}-\frac {p^2 x^2}{2}+1\right ) \]