67.4.17 problem Problem 3(c)

Internal problem ID [13982]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 3(c)
Date solved : Wednesday, March 05, 2025 at 10:24:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=24 \sin \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )+\operatorname {Heaviside}\left (t -\pi \right )\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: 9.366 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)+9*y(t) = 24*sin(t)*(Heaviside(t)+Heaviside(t-Pi)); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = 4 \left (1+\operatorname {Heaviside}\left (t -\pi \right )\right ) \sin \left (t \right )^{3} \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 24

ode=D[y[t],{t,2}]+9*y[t]==24*Sin[t]*(UnitStep[t]+UnitStep[t-Pi]); 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to 4 (\theta (\pi -t) (\theta (t)-2)+2) \sin ^3(t) \]

✓ Sympy. Time used: 5.458 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-24*(Heaviside(t) + Heaviside(t - pi))*sin(t) + 9*y(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)
 

\[ y{\left (t \right )} = \left (4 \sin ^{2}{\left (t \right )} \theta \left (t\right ) + 4 \sin ^{2}{\left (t \right )} \theta \left (t - \pi \right )\right ) \sin {\left (3 t \right )} - 8 \sin ^{3}{\left (t \right )} \cos {\left (2 t \right )} \theta \left (t\right ) - 8 \sin ^{3}{\left (t \right )} \cos {\left (2 t \right )} \theta \left (t - \pi \right ) \]