Internal problem ID [11520]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page
156
Problem number: 14.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_linear, `class A`]]
\[ \boxed {x^{\prime }+x=\operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (-2+t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 5.11 (sec). Leaf size: 40
dsolve([diff(x(t),t)=-x(t)+Heaviside(t-1)-Heaviside(t-2),x(0) = 1],x(t), singsol=all)
\[ x \left (t \right ) = \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2-t}-\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1}+{\mathrm e}^{-t}-\operatorname {Heaviside}\left (t -2\right )+\operatorname {Heaviside}\left (t -1\right ) \]
✓ Solution by Mathematica
Time used: 0.104 (sec). Leaf size: 48
DSolve[{x'[t]==-x[t]+UnitStep[t-1]-UnitStep[t-2],{x[0]==1}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} & t\leq 1 \\ e^{-t} \left (1-e+e^2\right ) & t>2 \\ e^{-t} \left (1-e+e^t\right ) & \text {True} \\ \end {array} \\ \end {array} \]