Internal problem ID [1795]
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: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\sin \relax (t ) y^{\prime \prime }+\cos \relax (t ) y^{\prime }+\frac {y}{t}=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 45
Order:=6; dsolve(sin(t)*diff(y(t),t$2)+cos(t)*diff(y(t),t)+1/t*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} t^{-i} \left (1+\left (\frac {1}{48}-\frac {i}{16}\right ) t^{2}+\left (\frac {1}{57600}-\frac {217 i}{57600}\right ) t^{4}+\mathrm {O}\left (t^{6}\right )\right )+c_{2} t^{i} \left (1+\left (\frac {1}{48}+\frac {i}{16}\right ) t^{2}+\left (\frac {1}{57600}+\frac {217 i}{57600}\right ) t^{4}+\mathrm {O}\left (t^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.043 (sec). Leaf size: 70
AsymptoticDSolveValue[Sin[t]*y''[t]+Cos[t]*y'[t]+1/t*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to \left (\frac {1}{19200}+\frac {i}{57600}\right ) c_1 t^i \left ((22+65 i) t^4+(720+960 i) t^2+(17280-5760 i)\right )-\left (\frac {1}{57600}+\frac {i}{19200}\right ) c_2 t^{-i} \left ((65+22 i) t^4+(960+720 i) t^2-(5760-17280 i)\right ) \]