Internal problem ID [1816]
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: 24.
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}-v^{2}\right ) 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: 81
Order:=6; dsolve(t^2*diff(y(t),t$2)+t*diff(y(t),t)+(t^2-v^2)*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} t^{-v} \left (1+\frac {1}{4 v -4} t^{2}+\frac {1}{32} \frac {1}{\left (v -2\right ) \left (v -1\right )} t^{4}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} t^{v} \left (1-\frac {1}{4 v +4} t^{2}+\frac {1}{32} \frac {1}{\left (v +2\right ) \left (v +1\right )} t^{4}+\mathrm {O}\left (t^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 160
AsymptoticDSolveValue[t^2*y''[t]+t*y'[t]+(t^2-v^2)*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_2 \left (\frac {t^4}{\left (-v^2-v+(1-v) (2-v)+2\right ) \left (-v^2-v+(3-v) (4-v)+4\right )}-\frac {t^2}{-v^2-v+(1-v) (2-v)+2}+1\right ) t^{-v}+c_1 \left (\frac {t^4}{\left (-v^2+v+(v+1) (v+2)+2\right ) \left (-v^2+v+(v+3) (v+4)+4\right )}-\frac {t^2}{-v^2+v+(v+1) (v+2)+2}+1\right ) t^v \]