90.26.7 problem 29

Internal problem ID [25408]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 379
Problem number : 29
Date solved : Friday, October 03, 2025 at 12:01:14 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\left \{\begin {array}{cc} t & 0\le t <1 \\ 0 & 1\le t <\infty \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.139 (sec). Leaf size: 45

ode:=diff(diff(y(t),t),t)-y(t) = piecewise(0 <= t and t < 1,t,1 <= t and t < infinity,0); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left \{\begin {array}{cc} 2 \sinh \left (t \right )-t & t <1 \\ 2 \sinh \left (1\right )-2 & t =1 \\ 2 \sinh \left (t \right )-\cosh \left (t -1\right )-\sinh \left (t -1\right ) & 1<t \end {array}\right . \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 63

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} e^{-t} \left (-1+e^{2 t}\right ) & t\leq 0 \\ -e^{t-1}-e^{-t}+e^t & t>1 \\ -t-e^{-t}+e^t & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.293 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((t, (t >= 0) & (t < 1)), (0, (t >= 1) & (t < oo))) - y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} - t & \text {for}\: t \geq 0 \wedge t < 1 \\0 & \text {for}\: t \geq 1 \wedge t < \infty \\\text {NaN} & \text {otherwise} \end {cases} + e^{t} - e^{- t} \]