67.21.5 problem 30.6 (e)

Internal problem ID [16911]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 30. Piecewise-defined functions and periodic functions. Additional Exercises. page 553
Problem number : 30.6 (e)
Date solved : Thursday, October 02, 2025 at 01:40:17 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\operatorname {Heaviside}\left (t -10\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: 0.126 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)+9*y(t) = Heaviside(t-10); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -10\right ) \left (\cos \left (-30+3 t \right )-1\right )}{9} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 26

ode=D[y[t],{t,2}]+9*y[t]==UnitStep[t-10]; 
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}{9} \sin ^2\left (15-\frac {3 t}{2}\right ) & t>10 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.473 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) - Heaviside(t - 10) + 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)
 

\[ y{\left (t \right )} = - \frac {\cos {\left (3 t - 30 \right )} \theta \left (t - 10\right )}{9} + \frac {\theta \left (t - 10\right )}{9} \]