76.19.13 problem 13

Internal problem ID [17658]
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 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 13
Date solved : Thursday, March 13, 2025 at 10:46:04 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-9 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=-3\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 9.426 (sec). Leaf size: 10

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-9*y(t) = 0; 
ic:=y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = -3, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \cos \left (\sqrt {3}\, t \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 13

ode=D[y[t],{t,4}]-9*y[t]==0; 
ic={y[0]==1,Derivative[1][y][0] == 0,Derivative[2][y][0] == -3,Derivative[3][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \cos \left (\sqrt {3} t\right ) \]

✓ Sympy. Time used: 0.144 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-9*y(t) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): -3, Subs(Derivative(y(t), (t, 3)), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \cos {\left (\sqrt {3} t \right )} \]