11.3.7 problem 14

Internal problem ID [1489]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.2, The Laplace Transform. Solution of Initial Value Problems. page 255
Problem number : 14
Date solved : Saturday, March 29, 2025 at 10:56:18 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-4 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 )&=1\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.139 (sec). Leaf size: 21

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-4*y(t) = 0; 
ic:=y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = 1, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\cos \left (\sqrt {2}\, t \right )}{4}+\frac {3 \cosh \left (\sqrt {2}\, t \right )}{4} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 43

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

\[ y(t)\to \frac {1}{8} \left (3 e^{-\sqrt {2} t}+3 e^{\sqrt {2} t}+2 \cos \left (\sqrt {2} t\right )\right ) \]

✓ Sympy. Time used: 0.160 (sec). Leaf size: 36

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

\[ y{\left (t \right )} = \frac {3 e^{\sqrt {2} t}}{8} + \frac {\cos {\left (\sqrt {2} t \right )}}{4} + \frac {3 e^{- \sqrt {2} t}}{8} \]