Internal problem ID [1815]
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: 23.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Lienard]
Solve \begin {gather*} \boxed {t^{2} y^{\prime \prime }+t y^{\prime }+t^{2} y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 41
Order:=6; dsolve(t^2*diff(y(t),t$2)+t*diff(y(t),t)+t^2*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \left (c_{2} \ln \relax (t )+c_{1}\right ) \left (1-\frac {1}{4} t^{2}+\frac {1}{64} t^{4}+\mathrm {O}\left (t^{6}\right )\right )+\left (\frac {1}{4} t^{2}-\frac {3}{128} t^{4}+\mathrm {O}\left (t^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 60
AsymptoticDSolveValue[t^2*y''[t]+t*y'[t]+t^2*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {t^4}{64}-\frac {t^2}{4}+1\right )+c_2 \left (-\frac {3 t^4}{128}+\frac {t^2}{4}+\left (\frac {t^4}{64}-\frac {t^2}{4}+1\right ) \log (t)\right ) \]