74.19.6 problem 6

Internal problem ID [16549]
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 : 6
Date solved : Thursday, March 13, 2025 at 08:19:23 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+4 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 15

ode:=diff(diff(x(t),t),t)+4*x(t) = 0; 
ic:=x(0) = 1, D(x)(0) = 1; 
dsolve([ode,ic],x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 15

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

\[ x(t)\to \cos (2 t)+\sin (t) \cos (t) \]

✓ Sympy. Time used: 0.060 (sec). Leaf size: 14

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

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