68.11.56 problem 69

Internal problem ID [17593]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 69
Date solved : Thursday, October 02, 2025 at 02:26:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+9 x&=\sin \left (3 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.019 (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), singsol=all);
 

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 93

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 \sin (3 t) \left (\int _1^t\frac {1}{6} \sin (6 K[2])dK[2]-\int _1^0\frac {1}{6} \sin (6 K[2])dK[2]\right )-\cos (3 t) \int _1^0-\frac {1}{3} \sin ^2(3 K[1])dK[1]+\cos (3 t) \int _1^t-\frac {1}{3} \sin ^2(3 K[1])dK[1] \end{align*}

✓ Sympy. Time used: 0.070 (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)
 

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