90.27.10 problem 10

Internal problem ID [25420]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 425
Problem number : 10
Date solved : Friday, October 03, 2025 at 12:01:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\operatorname {Heaviside}\left (t -2 \pi \right ) \sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.170 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 30

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (3 t) & t\leq 2 \pi \\ \frac {\sin ^3(t)}{6}+\cos (3 t) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 1.382 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) - sin(t)*Heaviside(t - 2*pi) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {\sin {\left (t \right )} \theta \left (t - 2 \pi \right )}{8} - \frac {\sin {\left (3 t \right )} \theta \left (t - 2 \pi \right )}{24} + \cos {\left (3 t \right )} \]