74.19.2 problem 2

Internal problem ID [16545]
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 : 2
Date solved : Thursday, March 13, 2025 at 08:17:37 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 9 x^{\prime \prime }+4 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 17

ode:=9*diff(diff(x(t),t),t)+4*x(t) = 0; 
ic:=x(0) = -1/2, D(x)(0) = 1; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x \left (t \right ) = \frac {3 \sin \left (\frac {2 t}{3}\right )}{2}-\frac {\cos \left (\frac {2 t}{3}\right )}{2} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 26

ode=9*D[x[t],{t,2}]+4*x[t]==0; 
ic={x[0]==-1/2,Derivative[1][x][0 ]==1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{2} \left (3 \sin \left (\frac {2 t}{3}\right )-\cos \left (\frac {2 t}{3}\right )\right ) \]

✓ Sympy. Time used: 0.066 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*x(t) + 9*Derivative(x(t), (t, 2)),0) 
ics = {x(0): -1/2, Subs(Derivative(x(t), t), t, 0): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {3 \sin {\left (\frac {2 t}{3} \right )}}{2} - \frac {\cos {\left (\frac {2 t}{3} \right )}}{2} \]