66.16.19 problem 19

Internal problem ID [16202]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 4. Forcing and Resonance. Section 4.1 page 399
Problem number : 19
Date solved : Thursday, October 02, 2025 at 10:43:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+y(t) = exp(-t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-t} \left (c_2 +t c_1 +\frac {1}{2} t^{2}\right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 27

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

\begin{align*} y(t)&\to \frac {1}{2} e^{-t} \left (t^2+2 c_2 t+2 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.131 (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 = {} 
dsolve(ode,func=y(t),ics=ics)
 

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