73.21.4 problem 30.6 (d)

Internal problem ID [15623]
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 (d)
Date solved : Tuesday, January 28, 2025 at 08:03:16 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 )&=4\\ y^{\prime }\left (0\right )&=6 \end{align*}

✓ Solution by Maple

Time used: 7.322 (sec). Leaf size: 20

dsolve([diff(y(t),t$2)=Heaviside(t-2),y(0) = 4, D(y)(0) = 6],y(t), singsol=all)
 

\[ y = 4+\frac {\operatorname {Heaviside}\left (t -2\right ) \left (t -2\right )^{2}}{2}+6 t \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 28

DSolve[{D[y[t],{t,2}]==UnitStep[t-2],{y[0]==4,Derivative[1][y][0] ==6}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 6 t+4 & t\leq 2 \\ \frac {t^2}{2}+4 t+6 & \text {True} \\ \end {array} \\ \end {array} \]