7.15.2 problem 2

Internal problem ID [2469]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.3, The method of Frobenius. Equal roots, and roots differering by an integer. Page 223
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 05:36:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 46

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

\[ y = \left (\left (c_2 \ln \left (t \right )+c_1 \right ) \left (1+t +\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{24} t^{4}+\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (-t -\frac {3}{4} t^{2}-\frac {11}{36} t^{3}-\frac {25}{288} t^{4}-\frac {137}{7200} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \right ) t \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 112

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

\[ y(t)\to c_1 t \left (\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2}+t+1\right )+c_2 \left (t \left (-\frac {137 t^5}{7200}-\frac {25 t^4}{288}-\frac {11 t^3}{36}-\frac {3 t^2}{4}-t\right )+t \left (\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2}+t+1\right ) \log (t)\right ) \]

✓ Sympy. Time used: 0.299 (sec). Leaf size: 27

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

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