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 : Sunday, March 30, 2025 at 12:12:35 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*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 36

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

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

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

ode=t^2*D[y[t],{t,2}]-t*D[y[t],t]-(t^2+5/4)*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

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

✓ Sympy. Time used: 0.870 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) - t*Derivative(y(t), t) - (t**2 + 5/4)*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} t^{\frac {5}{2}} \left (\frac {t^{2}}{10} + 1\right ) + \frac {C_{1} \left (- \frac {t^{4}}{8} - \frac {t^{2}}{2} + 1\right )}{\sqrt {t}} + O\left (t^{6}\right ) \]