Internal problem ID [2365]
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]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-y-\theta \left (t -1\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 33
dsolve([diff(y(t),t$2)-y(t)=Heaviside(t-1),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = \frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{t}}{2}+\frac {\theta \left (t -1\right ) \left ({\mathrm e}^{t -1}+{\mathrm e}^{-t +1}-2\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 23
DSolve[{y''[t]-y[t]==UnitStep[t-1],{y[0]==1,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} \cosh (t) & t\leq 1 \\ \cosh (1-t)+\cosh (t)-1 & \text {True} \\ \\ \\ \\ \\ \end{align*}