Internal problem ID [1809]
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: 17.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {t^{2} y^{\prime \prime }+t \left (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: 0.016 (sec). Leaf size: 45
Order:=6; dsolve(t^2*diff(y(t),t$2)+t*(t+1)*diff(y(t),t)-y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} t \left (1-\frac {1}{3} t +\frac {1}{12} t^{2}-\frac {1}{60} t^{3}+\frac {1}{360} t^{4}-\frac {1}{2520} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+\frac {c_{2} \left (-2+2 t -t^{2}+\frac {1}{3} t^{3}-\frac {1}{12} t^{4}+\frac {1}{60} t^{5}+\mathrm {O}\left (t^{6}\right )\right )}{t} \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 64
AsymptoticDSolveValue[t^2*y''[t]+t*(t+1)*y'[t]-y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {t^3}{24}-\frac {t^2}{6}+\frac {t}{2}+\frac {1}{t}-1\right )+c_2 \left (\frac {t^5}{360}-\frac {t^4}{60}+\frac {t^3}{12}-\frac {t^2}{3}+t\right ) \]