14.14.6 problem 6

Internal problem ID [2643]
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 : 6
Date solved : Monday, January 27, 2025 at 06:05:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{3} y^{\prime \prime }+\sin \left (t^{2}\right ) y^{\prime }+y t&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.089 (sec). Leaf size: 31

Order:=6; 
dsolve(t^3*diff(y(t),t$2)+sin(t^2)*diff(y(t),t)+t*y(t)=0,y(t),type='series',t=0);
 

\[ y = c_1 \,t^{-i} \left (1+\left (\frac {1}{240}-\frac {i}{120}\right ) t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \,t^{i} \left (1+\left (\frac {1}{240}+\frac {i}{120}\right ) t^{4}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 68

AsymptoticDSolveValue[t^3*D[y[t],{t,2}]+Sin[t^2]*D[y[t],t]+t*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to c_1 t^{-i} \left (\left (\frac {1}{23040}-\frac {i}{11520}\right ) t^8+\left (\frac {1}{240}-\frac {i}{120}\right ) t^4+1\right )+c_2 t^i \left (\left (\frac {1}{23040}+\frac {i}{11520}\right ) t^8+\left (\frac {1}{240}+\frac {i}{120}\right ) t^4+1\right ) \]