Internal problem ID [1775]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8, Series solutions. Page 195
Problem number: 11.
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 (-t^{2}+1\right ) y^{\prime \prime }-t y^{\prime }+\alpha ^{2} y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 71
Order:=6; dsolve((1-t^2)*diff(y(t),t$2)-t*diff(y(t),t)+alpha^2*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \left (1-\frac {\alpha ^{2} t^{2}}{2}+\frac {\alpha ^{2} \left (\alpha ^{2}-4\right ) t^{4}}{24}\right ) y \relax (0)+\left (t -\frac {\left (\alpha ^{2}-1\right ) t^{3}}{6}+\frac {\left (\alpha ^{4}-10 \alpha ^{2}+9\right ) t^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (t^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 88
AsymptoticDSolveValue[(1-t^2)*y''[t]-t*y'[t]+\[Alpha]^2*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_2 \left (\frac {\alpha ^4 t^5}{120}-\frac {\alpha ^2 t^5}{12}+\frac {3 t^5}{40}-\frac {\alpha ^2 t^3}{6}+\frac {t^3}{6}+t\right )+c_1 \left (\frac {\alpha ^4 t^4}{24}-\frac {\alpha ^2 t^4}{6}-\frac {\alpha ^2 t^2}{2}+1\right ) \]