7.14.1 problem 1

Internal problem ID [2441]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 05:35:55 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*} 0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 60

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=0);
 

\[ y = c_1 t \left (1-\frac {1}{4} t -\frac {5}{96} t^{2}-\frac {13}{1152} t^{3}-\frac {199}{92160} t^{4}-\frac {1123}{5529600} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (\ln \left (t \right ) \left (-\frac {1}{4} t +\frac {1}{16} t^{2}+\frac {5}{384} t^{3}+\frac {13}{4608} t^{4}+\frac {199}{368640} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (1-\frac {1}{4} t -\frac {1}{8} t^{2}+\frac {5}{2304} t^{3}+\frac {79}{13824} t^{4}+\frac {62027}{22118400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 87

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,0,5}]
 

\[ y(t)\to c_1 \left (\frac {t \left (13 t^3+60 t^2+288 t-1152\right ) \log (t)}{4608}+\frac {98 t^4+285 t^3+432 t^2-6912 t+6912}{6912}\right )+c_2 \left (-\frac {199 t^5}{92160}-\frac {13 t^4}{1152}-\frac {5 t^3}{96}-\frac {t^2}{4}+t\right ) \]

✓ Sympy. Time used: 0.356 (sec). Leaf size: 10

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=0,n=6)
 

\[ y{\left (t \right )} = C_{2} t + C_{1} + O\left (t^{6}\right ) \]