70.19.10 problem 10

Internal problem ID [19020]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 10
Date solved : Thursday, October 02, 2025 at 03:37:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=18 \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=7 \\ y^{\prime }\left (0\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.088 (sec). Leaf size: 19

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+y(t) = 18*exp(-t); 
ic:=[y(0) = 7, D(y)(0) = -2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left (9 t^{2}+5 t +7\right ) {\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 21

ode=D[y[t],{t,2}]+2*D[y[t],t]+y[t]==18*Exp[-t]; 
ic={y[0]==7,Derivative[1][y][0] == -2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t} \left (9 t^2+5 t+7\right ) \end{align*}

✓ Sympy. Time used: 0.128 (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)) - 18*exp(-t),0) 
ics = {y(0): 7, Subs(Derivative(y(t), t), t, 0): -2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t \left (9 t + 5\right ) + 7\right ) e^{- t} \]