73.21.3 problem 30.6 (c)

Internal problem ID [15622]
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 (c)
Date solved : Tuesday, January 28, 2025 at 08:03:15 AM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime }&=\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 )&=0 \end{align*}

✓ Solution by Maple

Time used: 9.033 (sec). Leaf size: 15

dsolve([diff(y(t),t$2)=Heaviside(t-2),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 21

DSolve[{D[y[t],{t,2}]==UnitStep[t-2],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} (t-2)^2 & t>2 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \]