63.16.1 problem 7

Internal problem ID [13198]
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
Date solved : Tuesday, January 28, 2025 at 05:12:17 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }-4 x&=1-\operatorname {Heaviside}\left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} x \left (0\right )&=0\\ x^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 10.336 (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: 0.030 (sec). Leaf size: 54

DSolve[{D[x[t],{t,2}]-4*x[t]==1-UnitStep[t-1],{x[0]==0,Derivative[1][x][0 ]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {1}{8} e^{-2 (t+1)} \left (\left (e^2-e^{2 t}\right )^2 \theta (1-t)+\left (e^2-1\right ) \left (e^{4 t}-e^2\right )\right ) \]