Internal problem ID [1807]
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: 15.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Lienard]
Solve \begin {gather*} \boxed {t y^{\prime \prime }-\left (t^{2}+2\right ) y^{\prime }+y t=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: 32
Order:=6; dsolve(t*diff(y(t),t$2)-(t^2+2)*diff(y(t),t)+t*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} t^{3} \left (1+\frac {1}{5} t^{2}+\frac {1}{35} t^{4}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} \left (12+6 t^{2}+\frac {3}{2} t^{4}+\mathrm {O}\left (t^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 44
AsymptoticDSolveValue[t*y''[t]-(t^2+2)*y'[t]+t*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {t^4}{8}+\frac {t^2}{2}+1\right )+c_2 \left (\frac {t^7}{35}+\frac {t^5}{5}+t^3\right ) \]