Internal problem ID [10504]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY.
2015.
Section: Chapter 3, Laplace transform. Section 3.3 The convolution property. Exercises page
162
Problem number: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }-4 x-1+\operatorname {Heaviside}\left (-1+t \right )=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 48
dsolve([diff(x(t),t$2)-4*x(t)=1-Heaviside(t-1),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
\[ x \left (t \right ) = \frac {{\mathrm e}^{-2 t}}{8}+\frac {{\mathrm e}^{2 t}}{8}+\frac {\operatorname {Heaviside}\left (t -1\right )}{4}-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2}}{8}-\frac {1}{4}-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{2 t -2}}{8} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 36
DSolve[{x''[t]-4*x[t]==1-UnitStep[t-1],{x[0]==0,x'[0]==0}},x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to {cc} \{ & {cc} \frac {\sinh ^2(t)}{2} & t\leq 1 \\ \frac {1}{4} (\cosh (2 t)-\cosh (2-2 t)) & \text {True} \\ \\ \\ \\ \\ \end{align*}