68.18.38 problem 44

Internal problem ID [17882]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 44
Date solved : Thursday, October 02, 2025 at 02:29:12 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=6 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 24

ode:=diff(diff(y(t),t),t)+9*y(t) = sin(3*t); 
ic:=[y(0) = 6, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 102

ode=D[y[t],{t,2}]+9*y[t]==Sin[3*t]; 
ic={y[0]==6,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.070 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) - sin(3*t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 6, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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