67.18.7 problem 27.1 (g)

Internal problem ID [16878]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 27. Differentiation and the Laplace transform. Additional Exercises. page 496
Problem number : 27.1 (g)
Date solved : Thursday, October 02, 2025 at 01:40:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=3 \operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.209 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+4*y(t) = 3*Heaviside(t-2); 
ic:=[y(0) = 0, D(y)(0) = 5]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {3 \operatorname {Heaviside}\left (t -2\right ) \sin \left (t -2\right )^{2}}{2}+\frac {5 \sin \left (2 t \right )}{2} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 37

ode=D[y[t],{t,2}]+4*y[t]==UnitStep[t-2]; 
ic={y[0]==0,Derivative[1][y][0] ==5}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} 5 \cos (t) \sin (t) & t\leq 2 \\ \frac {1}{4} (-\cos (4-2 t)+10 \sin (2 t)+1) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.531 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 3*Heaviside(t - 2) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 5} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {5 \sin {\left (2 t \right )}}{2} - \frac {3 \cos {\left (2 t - 4 \right )} \theta \left (t - 2\right )}{4} + \frac {3 \theta \left (t - 2\right )}{4} \]