38.2.9 problem 9

Internal problem ID [8227]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Section 1.2 Initial value problems. Exercises 1.2 at page 19
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 05:19:15 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} x \left (\frac {\pi }{6}\right )&={\frac {1}{2}} \\ x^{\prime }\left (\frac {\pi }{6}\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.074 (sec). Leaf size: 16

ode:=diff(diff(x(t),t),t)+x(t) = 0; 
ic:=[x(1/6*Pi) = 1/2, D(x)(1/6*Pi) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {\sin \left (t \right )}{4}+\frac {\sqrt {3}\, \cos \left (t \right )}{4} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 20

ode=D[x[t],{t,2}]+x[t]==0; 
ic={x[Pi/6]==1/2,Derivative[1][x][Pi/6] == 0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} \left (\sin (t)+\sqrt {3} \cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.040 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(pi/6): 1/2, Subs(Derivative(x(t), t), t, pi/6): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {\sin {\left (t \right )}}{4} + \frac {\sqrt {3} \cos {\left (t \right )}}{4} \]