20.23.11 problem Problem 11

Internal problem ID [3983]
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 11
Date solved : Tuesday, March 04, 2025 at 05:22:09 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=15 \sin \left (2 t \right )+\delta \left (t -\frac {\pi }{6}\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: 5.875 (sec). Leaf size: 29

ode:=diff(diff(y(t),t),t)+9*y(t) = 15*sin(2*t)+Dirac(t-1/6*Pi); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {\cos \left (3 t \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{6}\right )}{3}-2 \sin \left (3 t \right )+3 \sin \left (2 t \right ) \]

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 34

ode=D[y[t],{t,2}]+9*y[t]==15*Sin[2*t]+DiracDelta[t-Pi/6]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\frac {1}{3} \theta (6 t-\pi ) \cos (3 t)+3 \sin (2 t)-2 \sin (3 t) \]

✓ Sympy. Time used: 3.223 (sec). Leaf size: 117

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi/6) + 9*y(t) - 15*sin(2*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)
 

\[ y{\left (t \right )} = \left (- \frac {\int \left (\operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} + 15 \sin {\left (2 t \right )}\right ) \sin {\left (3 t \right )}\, dt}{3} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} \sin {\left (3 t \right )}\, dt}{3} + \frac {\int \limits ^{0} 15 \sin {\left (2 t \right )} \sin {\left (3 t \right )}\, dt}{3}\right ) \cos {\left (3 t \right )} + \left (\frac {\int \left (\operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} + 15 \sin {\left (2 t \right )}\right ) \cos {\left (3 t \right )}\, dt}{3} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} \cos {\left (3 t \right )}\, dt}{3} - \frac {\int \limits ^{0} 15 \sin {\left (2 t \right )} \cos {\left (3 t \right )}\, dt}{3}\right ) \sin {\left (3 t \right )} \]