1.18.2 problem 2

Internal problem ID [531]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.2 (Transformation of initial value problems). Problems at page 287
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 04:01:03 AM
CAS classification : [[_2nd_order, _missing_x]]

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

Using Laplace method With initial conditions

\begin{align*} x \left (0\right )&=3 \\ x^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.126 (sec). Leaf size: 17

ode:=diff(diff(x(t),t),t)+9*x(t) = 0; 
ic:=[x(0) = 3, D(x)(0) = 4]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

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

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

ode=D[x[t],{t,2}]+9*x[t]==0; 
ic={x[0]==3,Derivative[1][x][0] ==4}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.038 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(9*x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 3, Subs(Derivative(x(t), t), t, 0): 4} 
dsolve(ode,func=x(t),ics=ics)
 

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