20.22.8 problem Problem 34

Internal problem ID [3963]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.7. page 704
Problem number : Problem 34
Date solved : Tuesday, March 04, 2025 at 05:21:32 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\operatorname {Heaviside}\left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 2.555 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)-y(t) = Heaviside(t-1); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \cosh \left (t \right )+\operatorname {Heaviside}\left (-1+t \right ) \left (-1+\cosh \left (-1+t \right )\right ) \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 57

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

\[ y(t)\to \frac {1}{2} e^{-t-1} \left (\left (e-e^t\right )^2 (-\theta (1-t))+e^{2 t}-2 e^{t+1}+e^{2 t+1}+e^2+e\right ) \]

✓ Sympy. Time used: 0.567 (sec). Leaf size: 46

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

\[ y{\left (t \right )} = \left (\frac {e \theta \left (t - 1\right )}{2} + \frac {1}{2}\right ) e^{- t} + \left (\frac {\theta \left (t - 1\right )}{2 e} + \frac {1}{2}\right ) e^{t} - \theta \left (t - 1\right ) \]