Internal problem ID [1774]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8, Series solutions. Page 195
Problem number: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
Solve \begin {gather*} \boxed {\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+\alpha \left (\alpha +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 101
Order:=6; dsolve((1-t^2)*diff(y(t),t$2)-2*t*diff(y(t),t)+alpha*(alpha+1)*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \left (1-\frac {\alpha \left (1+\alpha \right ) t^{2}}{2}+\frac {\alpha \left (\alpha ^{3}+2 \alpha ^{2}-5 \alpha -6\right ) t^{4}}{24}\right ) y \relax (0)+\left (t +\frac {\left (-\alpha ^{2}-\alpha +2\right ) t^{3}}{6}+\frac {\left (\alpha ^{4}+2 \alpha ^{3}-13 \alpha ^{2}-14 \alpha +24\right ) t^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (t^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 127
AsymptoticDSolveValue[(1-t^2)*y''[t]-2*t*y'[t]+\[Alpha]*(\[Alpha]+1)*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_2 \left (\frac {1}{60} \left (-\alpha ^2-\alpha \right ) t^5-\frac {1}{120} \left (-\alpha ^2-\alpha \right ) \left (\alpha ^2+\alpha \right ) t^5-\frac {1}{10} \left (\alpha ^2+\alpha \right ) t^5+\frac {t^5}{5}-\frac {1}{6} \left (\alpha ^2+\alpha \right ) t^3+\frac {t^3}{3}+t\right )+c_1 \left (\frac {1}{24} \left (\alpha ^2+\alpha \right )^2 t^4-\frac {1}{4} \left (\alpha ^2+\alpha \right ) t^4-\frac {1}{2} \left (\alpha ^2+\alpha \right ) t^2+1\right ) \]