Internal problem ID [1814]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {t y^{\prime \prime }+\left (-t^{2}+1\right ) y^{\prime }+4 y t=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 41
Order:=6; dsolve(t*diff(y(t),t$2)+(1-t^2)*diff(y(t),t)+4*t*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \left (c_{2} \ln \relax (t )+c_{1}\right ) \left (1-t^{2}+\frac {1}{8} t^{4}+\mathrm {O}\left (t^{6}\right )\right )+\left (\frac {5}{4} t^{2}-\frac {9}{32} t^{4}+\mathrm {O}\left (t^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 56
AsymptoticDSolveValue[t*y''[t]+(1-t^2)*y'[t]+4*t*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {t^4}{8}-t^2+1\right )+c_2 \left (-\frac {9 t^4}{32}+\frac {5 t^2}{4}+\left (\frac {t^4}{8}-t^2+1\right ) \log (t)\right ) \]