22.3.8 problem 6.43

Internal problem ID [4521]
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 : 6.43
Date solved : Tuesday, September 30, 2025 at 07:33:49 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=54 t \,{\mathrm e}^{-2 t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.092 (sec). Leaf size: 24

ode:=diff(diff(y(t),t),t)+diff(y(t),t)-2*y(t) = 54*t*exp(-2*t); 
ic:=[y(0) = 6, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left (-9 t^{2}-6 t +6 \,{\mathrm e}^{3 t}\right ) {\mathrm e}^{-2 t} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 24

ode=D[y[t],{t,2}]+D[y[t],t]-2*y[t]==54*t*Exp[-2*t]; 
ic={y[0]==6,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 6 e^t-3 e^{-2 t} t (3 t+2) \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-54*t*exp(-2*t) - 2*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 6, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- 9 t^{2} - 6 t\right ) e^{- 2 t} + 6 e^{t} \]