Internal problem ID [1793]
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: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {t \left (t -2\right )^{2} y^{\prime \prime }+t 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: 60
Order:=6; dsolve(t*(t-2)^2*diff(y(t),t$2)+t*diff(y(t),t)+y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} t \left (1-\frac {1}{4} t -\frac {5}{96} t^{2}-\frac {13}{1152} t^{3}-\frac {199}{92160} t^{4}-\frac {1123}{5529600} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} \left (\ln \relax (t ) \left (-\frac {1}{4} t +\frac {1}{16} t^{2}+\frac {5}{384} t^{3}+\frac {13}{4608} t^{4}+\frac {199}{368640} t^{5}+\mathrm {O}\left (t^{6}\right )\right )+\left (1-\frac {1}{4} t -\frac {1}{8} t^{2}+\frac {5}{2304} t^{3}+\frac {79}{13824} t^{4}+\frac {62027}{22118400} t^{5}+\mathrm {O}\left (t^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.064 (sec). Leaf size: 87
AsymptoticDSolveValue[t*(t-2)^2*y''[t]+t*y'[t]+y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {t \left (13 t^3+60 t^2+288 t-1152\right ) \log (t)}{4608}+\frac {98 t^4+285 t^3+432 t^2-6912 t+6912}{6912}\right )+c_2 \left (-\frac {199 t^5}{92160}-\frac {13 t^4}{1152}-\frac {5 t^3}{96}-\frac {t^2}{4}+t\right ) \]