Internal problem ID [2874]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-y=\operatorname {Heaviside}\left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 2.203 (sec). Leaf size: 21
dsolve([diff(y(t),t$2)-y(t)=Heaviside(t-1),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \cosh \left (t \right )+\operatorname {Heaviside}\left (t -1\right ) \left (-1+\cosh \left (t -1\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 57
DSolve[{y''[t]-y[t]==UnitStep[t-1],{y[0]==1,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {1}{2} e^{-t-1} \left (\left (e-e^t\right )^2 (-\theta (1-t))+e^{2 t}-2 e^{t+1}+e^{2 t+1}+e^2+e\right ) \]