14.21.6 problem 6

Internal problem ID [2715]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.15, Higher order equations. Excercises page 263
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 02:40:14 PM
CAS classification : [[_high_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 19

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-y(t) = 0; 
ic:=y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = -1; 
dsolve([ode,ic],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 19

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

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

✓ Sympy. Time used: 0.111 (sec). Leaf size: 19

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

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