14.10.6 problem 6

Internal problem ID [2588]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.4. The method of variation of parameters. Excercises page 156
Problem number : 6
Date solved : Sunday, March 30, 2025 at 12:10:54 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=t^{{5}/{2}} {\mathrm e}^{-2 t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 13

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+4*y(t) = t^(5/2)*exp(-2*t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {4 t^{{9}/{2}} {\mathrm e}^{-2 t}}{63} \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 19

ode=D[y[t],{t,2}]+4*D[y[t],t]+4*y[t]==t^(5/2)*Exp[-2*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {4}{63} e^{-2 t} t^{9/2} \]

✓ Sympy. Time used: 0.310 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**(5/2)*exp(-2*t) + 4*y(t) + 4*Derivative(y(t), 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)
 

\[ y{\left (t \right )} = \frac {4 t^{\frac {9}{2}} e^{- 2 t}}{63} \]