13.14.15 problem 15

Internal problem ID [2455]
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 : 15
Date solved : Sunday, March 30, 2025 at 12:01:56 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 32

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

\[ y = c_1 \,t^{3} \left (1+\frac {1}{5} t^{2}+\frac {1}{35} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (12+6 t^{2}+\frac {3}{2} t^{4}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 44

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

\[ y(t)\to c_1 \left (\frac {t^4}{8}+\frac {t^2}{2}+1\right )+c_2 \left (\frac {t^7}{35}+\frac {t^5}{5}+t^3\right ) \]

✓ Sympy. Time used: 0.899 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{4}}{8} + \frac {t^{2}}{2} + 1\right ) + C_{1} t^{3} \left (\frac {t^{2}}{5} + 1\right ) + O\left (t^{6}\right ) \]