20.21.22 problem Problem 22

Internal problem ID [3949]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.4. page 689
Problem number : Problem 22
Date solved : Tuesday, March 04, 2025 at 05:19:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+4 y&=20 \sin \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 2.605 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+5*diff(y(t),t)+4*y(t) = 20*sin(2*t); 
ic:=y(0) = -1, D(y)(0) = 2; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 27

ode=D[y[t],{t,2}]+5*D[y[t],t]+4*y[t]==20*Sin[2*t]; 
ic={y[0]==-1,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-4 t} \left (2 e^{3 t}-1\right )-2 \cos (2 t) \]

✓ Sympy. Time used: 0.227 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 20*sin(2*t) + 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

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