73.21.3 problem 30.6 (c)

Internal problem ID [15543]
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 : Thursday, March 13, 2025 at 06:11:10 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*}

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

ode:=diff(diff(y(t),t),t) = Heaviside(t-2); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 0.415 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Heaviside(t - 2) + 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)
 

\[ y{\left (t \right )} = t \left (\begin {cases} 0 & \text {for}\: \left |{t}\right | < 2 \\t {G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {\frac {t}{2}} \right )} & \text {otherwise} \end {cases}\right ) - \begin {cases} 0 & \text {for}\: \left |{t}\right | < 2 \\t^{2} {G_{2, 2}^{0, 2}\left (\begin {matrix} -1, 1 & \\ & -2, 0 \end {matrix} \middle | {\frac {t}{2}} \right )} & \text {otherwise} \end {cases} \]