7.14.14 problem 14

Internal problem ID [2454]
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 : 14
Date solved : Tuesday, September 30, 2025 at 05:36:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} y^{\prime \prime }+\left (-t^{2}+t \right ) 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: 45

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

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 64

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

\[ y(t)\to c_1 \left (\frac {t^3}{24}+\frac {t^2}{6}+\frac {t}{2}+\frac {1}{t}+1\right )+c_2 \left (\frac {t^5}{360}+\frac {t^4}{60}+\frac {t^3}{12}+\frac {t^2}{3}+t\right ) \]

✓ Sympy. Time used: 0.298 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + (-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)
 

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