57.15.8 problem 6(h)

Internal problem ID [14472]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page 156
Problem number : 6(h)
Date solved : Thursday, October 02, 2025 at 09:37:37 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+9 x&=\sin \left (3 t \right ) \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: 0.095 (sec). Leaf size: 18

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

\[ x = \frac {\sin \left (3 t \right )}{18}-\frac {\cos \left (3 t \right ) t}{6} \]

✓ Mathematica. Time used: 0.067 (sec). Leaf size: 21

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

\begin{align*} x(t)&\to \frac {1}{18} (\sin (3 t)-3 t \cos (3 t)) \end{align*}

✓ Sympy. Time used: 0.071 (sec). Leaf size: 17

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

\[ x{\left (t \right )} = - \frac {t \cos {\left (3 t \right )}}{6} + \frac {\sin {\left (3 t \right )}}{18} \]