12.18.23 problem section 9.2, problem 23

Internal problem ID [2137]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.2. constant coefficient. Page 483
Problem number : section 9.2, problem 23
Date solved : Tuesday, March 04, 2025 at 01:50:42 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-16 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 29

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-16*y(x) = 0; 
ic:=y(0) = 2, D(y)(0) = 2, (D@@2)(y)(0) = -2, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-2 x}}{8}+\frac {5 \,{\mathrm e}^{2 x}}{8}+\frac {\sin \left (2 x \right )}{2}+\frac {5 \cos \left (2 x \right )}{4} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 34

ode=D[y[x],{x,4}]-16*y[x]==0; 
ic={y[0]==2,Derivative[1][y][0] ==2,Derivative[2][y][0] ==-2,Derivative[3][y][0]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{8} \left (e^{-2 x}+5 e^{2 x}+4 \sin (2 x)+10 \cos (2 x)\right ) \]

✓ Sympy. Time used: 0.139 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-16*y(x) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 2, Subs(Derivative(y(x), (x, 2)), x, 0): -2, Subs(Derivative(y(x), (x, 3)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {5 e^{2 x}}{8} + \frac {\sin {\left (2 x \right )}}{2} + \frac {5 \cos {\left (2 x \right )}}{4} + \frac {e^{- 2 x}}{8} \]