22.3.2 problem 6.37

Internal problem ID [4515]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.37
Date solved : Tuesday, September 30, 2025 at 07:33:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=9 \,{\mathrm e}^{-2 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=-6 \\ \end{align*}

✓ Maple. Time used: 0.097 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)+diff(y(t),t)-2*y(t) = 9*exp(-2*t); 
ic:=[y(0) = 3, D(y)(0) = -6]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left ({\mathrm e}^{3 t}-3 t +2\right ) {\mathrm e}^{-2 t} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 21

ode=D[y[t],{t,2}]+D[y[t],t]-2*y[t]==9*Exp[-2*t]; 
ic={y[0]==3,Derivative[1][y][0] == -6}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-2 t} \left (-3 t+e^{3 t}+2\right ) \end{align*}

✓ Sympy. Time used: 0.121 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 9*exp(-2*t),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): -6} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (2 - 3 t\right ) e^{- 2 t} + e^{t} \]