61.4.37 problem Problem 13(a)

Internal problem ID [15362]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 13(a)
Date solved : Thursday, October 02, 2025 at 10:12:07 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+y^{\prime \prime }+4 y^{\prime }+4 y&=8 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4 \\ y^{\prime }\left (0\right )&=-3 \\ y^{\prime \prime }\left (0\right )&=-3 \\ \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 20

ode:=diff(diff(diff(y(t),t),t),t)+diff(diff(y(t),t),t)+4*diff(y(t),t)+4*y(t) = 8; 
ic:=[y(0) = 4, D(y)(0) = -3, (D@@2)(y)(0) = -3]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 22

ode=D[ y[t],{t,3}]+D[y[t],{t,2}]+4*D[y[t],t]+4*y[t]==8; 
ic={y[0]==4,Derivative[1][y][0] ==-3,Derivative[2][y][0] ==-3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.127 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)) - 8,0) 
ics = {y(0): 4, Subs(Derivative(y(t), t), t, 0): -3, Subs(Derivative(y(t), (t, 2)), t, 0): -3} 
dsolve(ode,func=y(t),ics=ics)
 

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