14.12.12 problem 12 (b)

Internal problem ID [2622]
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 : 12 (b)
Date solved : Sunday, March 30, 2025 at 12:11:48 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+t^{3} y^{\prime }+3 t^{2} y&={\mathrm e}^{t} \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 16

Order:=6; 
ode:=diff(diff(y(t),t),t)+t^3*diff(y(t),t)+3*t^2*y(t) = exp(t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),type='series',t=0);
 

\[ y = \frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{24} t^{4}+\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 32

ode=D[y[t],{t,2}]+t^3*D[y[t],t]+3*t^2*y[t]==Exp[t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to \frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2} \]

✗ Sympy

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

ValueError : ODE t**3*Derivative(y(t), t) + 3*t**2*y(t) - exp(t) + Derivative(y(t), (t, 2)) does not match hint 2nd_power_series_regular