Internal problem ID [1800]
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: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre]
Solve \begin {gather*} \boxed {2 t y^{\prime \prime }+\left (-2 t +1\right ) y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 44
Order:=6; dsolve(2*t*diff(y(t),t$2)+(1-2*t)*diff(y(t),t)-y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} \sqrt {t}\, \left (1+\frac {2}{3} t +\frac {4}{15} t^{2}+\frac {8}{105} t^{3}+\frac {16}{945} t^{4}+\frac {32}{10395} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} \left (1+t +\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{24} t^{4}+\frac {1}{120} t^{5}+\mathrm {O}\left (t^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 81
AsymptoticDSolveValue[2*t*y''[t]+(1-2*t)*y'[t]-y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \sqrt {t} \left (\frac {32 t^5}{10395}+\frac {16 t^4}{945}+\frac {8 t^3}{105}+\frac {4 t^2}{15}+\frac {2 t}{3}+1\right )+c_2 \left (\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2}+t+1\right ) \]