14.14.13 problem 13

Internal problem ID [2650]
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 : 13
Date solved : Monday, January 27, 2025 at 06:05:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} y^{\prime \prime }-t y^{\prime }-\left (t^{2}+\frac {5}{4}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 36

Order:=6; 
dsolve(t^2*diff(y(t),t$2)-t*diff(y(t),t)-(t^2+5/4)*y(t)=0,y(t),type='series',t=0);
 

\[ y = \frac {c_1 \,t^{3} \left (1+\frac {1}{10} t^{2}+\frac {1}{280} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (12-6 t^{2}-\frac {3}{2} t^{4}+\operatorname {O}\left (t^{6}\right )\right )}{\sqrt {t}} \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 58

AsymptoticDSolveValue[t^2*D[y[t],{t,2}]-t*D[y[t],t]-(t^2+5/4)*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to c_1 \left (-\frac {t^{7/2}}{8}-\frac {t^{3/2}}{2}+\frac {1}{\sqrt {t}}\right )+c_2 \left (\frac {t^{13/2}}{280}+\frac {t^{9/2}}{10}+t^{5/2}\right ) \]