68.19.8 problem 8

Internal problem ID [17917]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 5. Applications of Higher Order Equations. Exercises 5.1, page 232
Problem number : 8
Date solved : Thursday, October 02, 2025 at 02:29:44 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+256 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 17

ode:=diff(diff(x(t),t),t)+256*x(t) = 0; 
ic:=[x(0) = 2, D(x)(0) = 4]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {\sin \left (16 t \right )}{4}+2 \cos \left (16 t \right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 20

ode=D[x[t],{t,2}]+256*x[t]==0; 
ic={x[0]==2,Derivative[1][x][0 ]==4}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} (\sin (16 t)+8 \cos (16 t)) \end{align*}

✓ Sympy. Time used: 0.036 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(256*x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 2, Subs(Derivative(x(t), t), t, 0): 4} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {\sin {\left (16 t \right )}}{4} + 2 \cos {\left (16 t \right )} \]