Internal problem ID [1812]
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: 20.
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 y^{\prime }-\left (t +1\right ) 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: 63
Order:=6; dsolve(t^2*diff(y(t),t$2)+t*diff(y(t),t)-(1+t)*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \frac {c_{1} t^{2} \left (1+\frac {1}{3} t +\frac {1}{24} t^{2}+\frac {1}{360} t^{3}+\frac {1}{8640} t^{4}+\frac {1}{302400} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} \left (\ln \relax (t ) \left (t^{2}+\frac {1}{3} t^{3}+\frac {1}{24} t^{4}+\frac {1}{360} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+\left (-2+2 t -\frac {4}{9} t^{3}-\frac {25}{288} t^{4}-\frac {157}{21600} t^{5}+\mathrm {O}\left (t^{6}\right )\right )\right )}{t} \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 83
AsymptoticDSolveValue[t^2*y''[t]+t*y'[t]-(1+t)*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {31 t^4+176 t^3+144 t^2-576 t+576}{576 t}-\frac {1}{48} t \left (t^2+8 t+24\right ) \log (t)\right )+c_2 \left (\frac {t^5}{8640}+\frac {t^4}{360}+\frac {t^3}{24}+\frac {t^2}{3}+t\right ) \]