Internal problem ID [13615]
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: 33.9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\lambda y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 63
Order:=6; dsolve((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+lambda*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (1-\frac {\lambda \,x^{2}}{2}+\frac {\lambda \left (\lambda -4\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\lambda -1\right ) x^{3}}{6}+\frac {\left (\lambda -1\right ) \left (\lambda -9\right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 80
AsymptoticDSolveValue[(1-x^2)*y''[x]-x*y'[x]+\[Lambda]*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {\lambda ^2 x^5}{120}-\frac {\lambda x^5}{12}+\frac {3 x^5}{40}-\frac {\lambda x^3}{6}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {\lambda ^2 x^4}{24}-\frac {\lambda x^4}{6}-\frac {\lambda x^2}{2}+1\right ) \]