90.27.7 problem 7

Internal problem ID [25417]
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 : 7
Date solved : Friday, October 03, 2025 at 12:01:19 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\operatorname {Heaviside}\left (-3+t \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: 0.041 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)+9*y(t) = Heaviside(t-3); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -3\right ) \left (\cos \left (-9+3 t \right )-1\right )}{9} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 31

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {2}{3} \sin ^2\left (\frac {1}{2} \sqrt {3} (t-3)\right ) & t>3 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.534 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(3*y(t) - Heaviside(t - 3) + 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)
 

\[ y{\left (t \right )} = - \frac {\cos {\left (\sqrt {3} \left (t - 3\right ) \right )} \theta \left (t - 3\right )}{3} + \frac {\theta \left (t - 3\right )}{3} \]