38.2.10 problem 10

Internal problem ID [8228]
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 : 10
Date solved : Tuesday, September 30, 2025 at 05:19:17 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 }{4}\right )&=\sqrt {2} \\ x^{\prime }\left (\frac {\pi }{4}\right )&=2 \sqrt {2} \\ \end{align*}

✓ Maple. Time used: 0.080 (sec). Leaf size: 13

ode:=diff(diff(x(t),t),t)+x(t) = 0; 
ic:=[x(1/4*Pi) = 2^(1/2), D(x)(1/4*Pi) = 2*2^(1/2)]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = 3 \sin \left (t \right )-\cos \left (t \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 14

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

\begin{align*} x(t)&\to 3 \sin (t)-\cos (t) \end{align*}

✓ Sympy. Time used: 0.040 (sec). Leaf size: 10

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

\[ x{\left (t \right )} = 3 \sin {\left (t \right )} - \cos {\left (t \right )} \]