Internal problem ID [1811]
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: 19.
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 }+\left (t^{2}-3 t \right ) y^{\prime }+3 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: 61
Order:=6; dsolve(t^2*diff(y(t),t$2)+(t^2-3*t)*diff(y(t),t)+3*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \left (\left (1-t +\frac {1}{2} t^{2}-\frac {1}{6} t^{3}+\frac {1}{24} t^{4}-\frac {1}{120} t^{5}+\mathrm {O}\left (t^{6}\right )\right ) c_{1} t^{2}+c_{2} \left (\ln \relax (t ) \left (2 t^{2}-2 t^{3}+t^{4}-\frac {1}{3} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+\left (-2-2 t +3 t^{2}-t^{3}+\frac {1}{9} t^{5}+\mathrm {O}\left (t^{6}\right )\right )\right )\right ) t \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 76
AsymptoticDSolveValue[t^2*y''[t]+(t^2-3*t)*y'[t]+3*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {1}{4} t \left (t^4-4 t^2+4 t+4\right )-\frac {1}{2} t^3 \left (t^2-2 t+2\right ) \log (t)\right )+c_2 \left (\frac {t^7}{24}-\frac {t^6}{6}+\frac {t^5}{2}-t^4+t^3\right ) \]