7.15.3 problem 3

Internal problem ID [2470]
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 : 3
Date solved : Tuesday, September 30, 2025 at 05:36:20 AM
CAS classification : [_Bessel]

\begin{align*} t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-1\right ) 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*diff(y(t),t)+(t^2-1)*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \frac {c_1 \,t^{2} \left (1-\frac {1}{8} t^{2}+\frac {1}{192} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (\ln \left (t \right ) \left (t^{2}-\frac {1}{8} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+\left (-2+\frac {3}{32} t^{4}+\operatorname {O}\left (t^{6}\right )\right )\right )}{t} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 58

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

\[ y(t)\to c_2 \left (\frac {t^5}{192}-\frac {t^3}{8}+t\right )+c_1 \left (\frac {1}{16} t \left (t^2-8\right ) \log (t)-\frac {5 t^4-16 t^2-64}{64 t}\right ) \]

✓ Sympy. Time used: 0.266 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = C_{1} t \left (\frac {t^{4}}{192} - \frac {t^{2}}{8} + 1\right ) + O\left (t^{6}\right ) \]