Internal problem ID [1813]
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: 21.
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 }+t y^{\prime }+2 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: 60
Order:=6; dsolve(t*diff(y(t),t$2)+t*diff(y(t),t)+2*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} t \left (1-\frac {3}{2} t +t^{2}-\frac {5}{12} t^{3}+\frac {1}{8} t^{4}-\frac {7}{240} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} \left (\ln \relax (t ) \left (\left (-2\right ) t +3 t^{2}-2 t^{3}+\frac {5}{6} t^{4}-\frac {1}{4} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+\left (1-t -2 t^{2}+\frac {5}{2} t^{3}-\frac {49}{36} t^{4}+\frac {23}{48} t^{5}+\mathrm {O}\left (t^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 83
AsymptoticDSolveValue[t*y''[t]+t*y'[t]+2*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {1}{6} t \left (5 t^3-12 t^2+18 t-12\right ) \log (t)+\frac {1}{36} \left (-79 t^4+162 t^3-180 t^2+36 t+36\right )\right )+c_2 \left (\frac {t^5}{8}-\frac {5 t^4}{12}+t^3-\frac {3 t^2}{2}+t\right ) \]