14.23.12 problem Problem 12

Internal problem ID [3984]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.8. page 710
Problem number : Problem 12
Date solved : Tuesday, September 30, 2025 at 07:00:00 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=4 \cos \left (3 t \right )+\delta \left (t -\frac {\pi }{3}\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.405 (sec). Leaf size: 40

ode:=diff(diff(y(t),t),t)+16*y(t) = 4*cos(3*t)+Dirac(t-1/3*Pi); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {\left (\sqrt {3}\, \cos \left (4 t \right )-\sin \left (4 t \right )\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{3}\right )}{8}+\frac {4 \cos \left (3 t \right )}{7}-\frac {4 \cos \left (4 t \right )}{7} \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 50

ode=D[y[t],{t,2}]+16*y[t]==4*Cos[3*t]+DiracDelta[t-Pi/3]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{8} \theta (3 t-\pi ) \left (\sqrt {3} \cos (4 t)-\sin (4 t)\right )+\frac {4}{7} (\cos (3 t)-\cos (4 t)) \end{align*}

✓ Sympy. Time used: 3.435 (sec). Leaf size: 117

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi/3) + 16*y(t) - 4*cos(3*t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {\int \left (\operatorname {Dirac}{\left (t - \frac {\pi }{3} \right )} + 4 \cos {\left (3 t \right )}\right ) \sin {\left (4 t \right )}\, dt}{4} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{3} \right )} \sin {\left (4 t \right )}\, dt}{4} + \frac {\int \limits ^{0} 4 \sin {\left (4 t \right )} \cos {\left (3 t \right )}\, dt}{4}\right ) \cos {\left (4 t \right )} + \left (\frac {\int \left (\operatorname {Dirac}{\left (t - \frac {\pi }{3} \right )} + 4 \cos {\left (3 t \right )}\right ) \cos {\left (4 t \right )}\, dt}{4} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{3} \right )} \cos {\left (4 t \right )}\, dt}{4} - \frac {\int \limits ^{0} 4 \cos {\left (3 t \right )} \cos {\left (4 t \right )}\, dt}{4}\right ) \sin {\left (4 t \right )} \]