14.14.16 problem 16

Internal problem ID [2653]
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 : 16
Date solved : Tuesday, March 04, 2025 at 02:33:05 PM
CAS classification : [_Laguerre, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 44

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

\[ y = c_1 \left (1+\frac {1}{3} t +\frac {1}{12} t^{2}+\frac {1}{60} t^{3}+\frac {1}{360} t^{4}+\frac {1}{2520} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\frac {c_2 \left (-2-2 t -t^{2}-\frac {1}{3} t^{3}-\frac {1}{12} t^{4}-\frac {1}{60} t^{5}+\operatorname {O}\left (t^{6}\right )\right )}{t^{2}} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 60

ode=t^2*D[y[t],{t,2}]+(3*t-t^2)*D[y[t],t]-t*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \left (\frac {t^2}{24}+\frac {1}{t^2}+\frac {t}{6}+\frac {1}{t}+\frac {1}{2}\right )+c_2 \left (\frac {t^4}{360}+\frac {t^3}{60}+\frac {t^2}{12}+\frac {t}{3}+1\right ) \]

✓ Sympy. Time used: 0.997 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) - t*y(t) + (-t**2 + 3*t)*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{5}}{2520} + \frac {t^{4}}{360} + \frac {t^{3}}{60} + \frac {t^{2}}{12} + \frac {t}{3} + 1\right ) + \frac {C_{1} \left (t + 1\right )}{t^{2}} + O\left (t^{6}\right ) \]