22.3.1 problem 6.36

Internal problem ID [4514]
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 : 6.36
Date solved : Tuesday, September 30, 2025 at 07:33:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+3 y&=60 \cos \left (3 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.110 (sec). Leaf size: 29

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+3*y(t) = 60*cos(3*t); 
ic:=[y(0) = 1, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -2 \cos \left (3 t \right )+4 \sin \left (3 t \right )+5 \,{\mathrm e}^{-3 t}-2 \,{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 33

ode=D[y[t],{t,2}]+4*D[y[t],t]+3*y[t]==60*Cos[3*t]; 
ic={y[0]==1,Derivative[1][y][0] == -1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-3 t} \left (5-2 e^{2 t}\right )+4 \sin (3 t)-2 \cos (3 t) \end{align*}

✓ Sympy. Time used: 0.138 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(3*y(t) - 60*cos(3*t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 4 \sin {\left (3 t \right )} - 2 \cos {\left (3 t \right )} - 2 e^{- t} + 5 e^{- 3 t} \]