90.14.6 problem 15

Internal problem ID [25234]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 3. Second Order Constant Coefficient Linear Differential Equations. Exercises at page 243
Problem number : 15
Date solved : Thursday, October 02, 2025 at 11:59:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&={\mathrm e}^{-3 t} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+5*y(t) = exp(-3*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-2 t} \sin \left (t \right ) c_2 +{\mathrm e}^{-2 t} \cos \left (t \right ) c_1 +\frac {{\mathrm e}^{-3 t}}{2} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 34

ode=D[y[t],{t,2}]+4*D[y[t],{t,1}]+5*y[t]==Exp[-3*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{2} e^{-3 t} \left (2 c_2 e^t \cos (t)+2 c_1 e^t \sin (t)+1\right ) \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - exp(-3*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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