90.14.17 problem 26

Internal problem ID [25245]
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 : 26
Date solved : Thursday, October 02, 2025 at 11:59:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=8 \,{\mathrm e}^{-t} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=8 \\ \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 8*exp(-t); 
ic:=[y(0) = 0, D(y)(0) = 8]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 25

ode=D[y[t],{t,2}]+2*D[y[t],{t,1}]+5*y[t]==8*Exp[-t]; 
ic={y[0]==0,Derivative[1][y][0] ==8}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t} (4 \sin (2 t)-2 \cos (2 t)+2) \end{align*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 8*exp(-t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 8} 
dsolve(ode,func=y(t),ics=ics)
 

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