14.12.4 problem 4

Internal problem ID [2614]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8. Series solutions. Excercises page 197
Problem number : 4
Date solved : Tuesday, March 04, 2025 at 02:32:14 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }-t^{3} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

Order:=6; 
ode:=diff(diff(y(t),t),t)-t^3*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \left (1+\frac {t^{5}}{20}\right ) y \left (0\right )+t y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 20

ode=D[y[t],{t,2}]-t^3*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \left (\frac {t^5}{20}+1\right )+c_2 t \]

✓ Sympy. Time used: 0.701 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**3*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{5}}{20} + 1\right ) + C_{1} t \left (\frac {t^{5}}{30} + 1\right ) + O\left (t^{6}\right ) \]