Internal problem ID [11522]
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]]
\[ \boxed {x^{\prime \prime }-4 x=1-\operatorname {Heaviside}\left (t -1\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 5.437 (sec). Leaf size: 24
dsolve([diff(x(t),t$2)-4*x(t)=1-Heaviside(t-1),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
\[ x \left (t \right ) = -\frac {1}{4}+\frac {\cosh \left (2 t \right )}{4}-\frac {\operatorname {Heaviside}\left (t -1\right ) \sinh \left (t -1\right )^{2}}{2} \]
✓ Solution by Mathematica
Time used: 0.046 (sec). Leaf size: 54
DSolve[{x''[t]-4*x[t]==1-UnitStep[t-1],{x[0]==0,x'[0]==0}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {1}{8} e^{-2 (t+1)} \left (\left (e^2-e^{2 t}\right )^2 \theta (1-t)+\left (e^2-1\right ) \left (e^{4 t}-e^2\right )\right ) \]