8.11.5 problem 5

Internal problem ID [2598]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.5. Method of judicious guessing. Excercises page 164
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 05:47:38 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: 0.004 (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: 0.015 (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: 0.130 (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)
 

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