Internal problem ID [1822]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8.3, The method of Frobenius. Equal roots, and roots differering by an integer. Page
223
Problem number: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Bessel]
Solve \begin {gather*} \boxed {t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-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: 47
Order:=6; dsolve(t^2*diff(y(t),t$2)+t*diff(y(t),t)+(t^2-1)*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = \frac {c_{1} t^{2} \left (1-\frac {1}{8} t^{2}+\frac {1}{192} t^{4}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} \left (\ln \relax (t ) \left (t^{2}-\frac {1}{8} t^{4}+\mathrm {O}\left (t^{6}\right )\right )+\left (-2+\frac {3}{32} t^{4}+\mathrm {O}\left (t^{6}\right )\right )\right )}{t} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 58
AsymptoticDSolveValue[t^2*y''[t]+t*y'[t]+(t^2-1)*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_2 \left (\frac {t^5}{192}-\frac {t^3}{8}+t\right )+c_1 \left (\frac {1}{16} t \left (t^2-8\right ) \log (t)-\frac {5 t^4-16 t^2-64}{64 t}\right ) \]