73.18.7 problem 27.1 (g)

Internal problem ID [15591]
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 : Tuesday, January 28, 2025 at 08:02:55 AM
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*}

Solution by Maple

Time used: 9.179 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)+4*y(t)=3*Heaviside(t-2),y(0) = 0, D(y)(0) = 5],y(t), singsol=all)
 
\[ y = \frac {3 \operatorname {Heaviside}\left (t -2\right ) \sin \left (t -2\right )^{2}}{2}+\frac {5 \sin \left (2 t \right )}{2} \]

Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 37

DSolve[{D[y[t],{t,2}]+4*y[t]==UnitStep[t-2],{y[0]==0,Derivative[1][y][0] ==5}},y[t],t,IncludeSingularSolutions -> True]
 
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 5 \cos (t) \sin (t) & t\leq 2 \\ \frac {1}{4} (-\cos (4-2 t)+10 \sin (2 t)+1) & \text {True} \\ \end {array} \\ \end {array} \]