14.14.23 problem 23

Internal problem ID [2660]
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 : 23
Date solved : Tuesday, March 04, 2025 at 02:33:15 PM
CAS classification : [_Lienard]

\begin{align*} t^{2} y^{\prime \prime }+t y^{\prime }+t^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 32

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

\[ y = \left (c_2 \ln \left (t \right )+c_1 \right ) \left (1-\frac {1}{4} t^{2}+\frac {1}{64} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+\left (\frac {1}{4} t^{2}-\frac {3}{128} t^{4}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 60

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

\[ y(t)\to c_1 \left (\frac {t^4}{64}-\frac {t^2}{4}+1\right )+c_2 \left (-\frac {3 t^4}{128}+\frac {t^2}{4}+\left (\frac {t^4}{64}-\frac {t^2}{4}+1\right ) \log (t)\right ) \]

✓ Sympy. Time used: 0.796 (sec). Leaf size: 19

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

\[ y{\left (t \right )} = C_{1} \left (\frac {t^{4}}{64} - \frac {t^{2}}{4} + 1\right ) + O\left (t^{6}\right ) \]