Internal problem ID [1808]
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: 16.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {t^{2} y^{\prime \prime }+\left (-t^{2}+3 t \right ) y^{\prime }-y t=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: 44
Order:=6; dsolve(t^2*diff(y(t),t$2)+(3*t-t^2)*diff(y(t),t)-t*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} \left (1+\frac {1}{3} t +\frac {1}{12} t^{2}+\frac {1}{60} t^{3}+\frac {1}{360} t^{4}+\frac {1}{2520} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+\frac {c_{2} \left (-2-2 t -t^{2}-\frac {1}{3} t^{3}-\frac {1}{12} t^{4}-\frac {1}{60} t^{5}+\mathrm {O}\left (t^{6}\right )\right )}{t^{2}} \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 60
AsymptoticDSolveValue[t^2*y''[t]+(3*t-t^2)*y'[t]-t*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {t^2}{24}+\frac {1}{t^2}+\frac {t}{6}+\frac {1}{t}+\frac {1}{2}\right )+c_2 \left (\frac {t^4}{360}+\frac {t^3}{60}+\frac {t^2}{12}+\frac {t}{3}+1\right ) \]