20.21.23 problem Problem 23

Internal problem ID [3950]
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 23
Date solved : Tuesday, March 04, 2025 at 05:19:56 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: 2.321 (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');
 

\[ y = -\frac {{\mathrm e}^{-4 t}}{3}-2 \cos \left (2 t \right )+\frac {10 \,{\mathrm e}^{-t}}{3} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 30

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]
 

\[ y(t)\to \frac {1}{3} e^{-4 t} \left (10 e^{3 t}-1\right )-2 \cos (2 t) \]

✓ Sympy. Time used: 0.222 (sec). Leaf size: 24

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)
 

\[ y{\left (t \right )} = - 2 \cos {\left (2 t \right )} + \frac {10 e^{- t}}{3} - \frac {e^{- 4 t}}{3} \]