7.18.8 problem 8

Internal problem ID [537]
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 : 8
Date solved : Tuesday, March 04, 2025 at 11:26:20 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+9 x&=1 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.194 (sec). Leaf size: 12

ode:=diff(diff(x(t),t),t)+9*x(t) = 1; 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x \left (t \right ) = -\frac {\cos \left (3 t \right )}{9}+\frac {1}{9} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 17

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

\[ x(t)\to \frac {2}{9} \sin ^2\left (\frac {3 t}{2}\right ) \]

✓ Sympy. Time used: 0.089 (sec). Leaf size: 12

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

\[ x{\left (t \right )} = \frac {1}{9} - \frac {\cos {\left (3 t \right )}}{9} \]