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 : Monday, January 27, 2025 at 08:05:16 AM
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*}

✓ Solution by Maple

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

dsolve([diff(y(t),t$2)-y(t)=Heaviside(t-1),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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