14.14.3 problem 3

Internal problem ID [2640]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8.2, Regular singular points, the method of Frobenius. Excercises page 216
Problem number : 3
Date solved : Monday, January 27, 2025 at 06:05:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \sin \left (t \right ) y^{\prime \prime }+\cos \left (t \right ) y^{\prime }+\frac {y}{t}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.161 (sec). Leaf size: 35

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);
 

\[ y = c_1 \,t^{-i} \left (1+\left (\frac {1}{48}-\frac {i}{16}\right ) t^{2}+\left (\frac {1}{57600}-\frac {217 i}{57600}\right ) t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \,t^{i} \left (1+\left (\frac {1}{48}+\frac {i}{16}\right ) t^{2}+\left (\frac {1}{57600}+\frac {217 i}{57600}\right ) t^{4}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 70

AsymptoticDSolveValue[Sin[t]*D[y[t],{t,2}]+Cos[t]*D[y[t],t]+1/t*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to \left (\frac {1}{19200}+\frac {i}{57600}\right ) c_1 t^i \left ((22+65 i) t^4+(720+960 i) t^2+(17280-5760 i)\right )-\left (\frac {1}{57600}+\frac {i}{19200}\right ) c_2 t^{-i} \left ((65+22 i) t^4+(960+720 i) t^2-(5760-17280 i)\right ) \]