14.14.2 problem 2

Internal problem ID [2639]
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 : 2
Date solved : Sunday, March 30, 2025 at 12:12:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 2 \end{align*}

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 112

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

\[ y(t)\to c_2 e^{\frac {1}{t-2}} \left (\frac {247853}{240} (t-2)^5+\frac {4069}{24} (t-2)^4+\frac {199}{6} (t-2)^3+8 (t-2)^2+\frac {5 (t-2)}{2}+1\right ) (t-2)^2+c_1 \left (-\frac {641}{480} (t-2)^5+\frac {25}{48} (t-2)^4-\frac {7}{24} (t-2)^3+\frac {1}{4} (t-2)^2+\frac {2-t}{2}+1\right ) \]

✗ Sympy

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

ValueError : ODE t*(t - 2)**2*Derivative(y(t), (t, 2)) + t*Derivative(y(t), t) + y(t) does not match hint 2nd_power_series_regular