76.18.9 problem 20

Internal problem ID [17649]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 20
Date solved : Monday, March 31, 2025 at 04:23:36 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=-2 \end{align*}

✓ Maple. Time used: 0.127 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 17

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

\[ y(t)\to e^{-t}+\sin (t)+3 \cos (t) \]

✓ Sympy. Time used: 0.151 (sec). Leaf size: 15

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

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