14.14.20 problem 20

Internal problem ID [2657]
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 : 20
Date solved : Sunday, March 30, 2025 at 12:12:47 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} y^{\prime \prime }+t y^{\prime }-\left (1+t \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 62

Order:=6; 
ode:=t^2*diff(diff(y(t),t),t)+t*diff(y(t),t)-(t+1)*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \frac {c_1 \,t^{2} \left (1+\frac {1}{3} t +\frac {1}{24} t^{2}+\frac {1}{360} t^{3}+\frac {1}{8640} t^{4}+\frac {1}{302400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (\ln \left (t \right ) \left (t^{2}+\frac {1}{3} t^{3}+\frac {1}{24} t^{4}+\frac {1}{360} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (-2+2 t -\frac {4}{9} t^{3}-\frac {25}{288} t^{4}-\frac {157}{21600} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right )}{t} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 83

ode=t^2*D[y[t],{t,2}]+t*D[y[t],t]-(1+t)*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \left (\frac {31 t^4+176 t^3+144 t^2-576 t+576}{576 t}-\frac {1}{48} t \left (t^2+8 t+24\right ) \log (t)\right )+c_2 \left (\frac {t^5}{8640}+\frac {t^4}{360}+\frac {t^3}{24}+\frac {t^2}{3}+t\right ) \]

✓ Sympy. Time used: 0.803 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + t*Derivative(y(t), t) - (t + 1)*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{1} t \left (\frac {t^{4}}{8640} + \frac {t^{3}}{360} + \frac {t^{2}}{24} + \frac {t}{3} + 1\right ) + O\left (t^{6}\right ) \]