14.16.3 problem 17

Internal problem ID [2673]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.9, The method of Laplace transform. Excercises page 232
Problem number : 17
Date solved : Tuesday, March 04, 2025 at 02:33:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Maple. Time used: 0.822 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+y(t) = exp(-t); 
ic:=y(0) = 1, D(y)(0) = 3; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-t} \left (t^{2}+8 t +2\right )}{2} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 22

ode=D[y[t],{t,2}]+2*D[y[t],t]+y[t]==Exp[-t]; 
ic={y[0]==1,Derivative[1][y][0] ==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{2} e^{-t} \left (t^2+8 t+2\right ) \]

✓ Sympy. Time used: 0.228 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - exp(-t),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t \left (\frac {t}{2} + 4\right ) + 1\right ) e^{- t} \]