63.15.7 problem 6(g)

Internal problem ID [13190]
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 : 6(g)
Date solved : Tuesday, January 28, 2025 at 05:12:09 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+\frac {2 x^{\prime }}{5}+2 x&=1-\operatorname {Heaviside}\left (t -5\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: 13.320 (sec). Leaf size: 55

dsolve([diff(x(t),t$2)+4/10*diff(x(t),t)+2*x(t)=1-Heaviside(t-5),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {1}{2}+\left (\frac {1}{4}+\frac {i}{28}\right ) \operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{\left (-\frac {1}{5}-\frac {7 i}{5}\right ) \left (t -5\right )}+\left (\frac {1}{4}-\frac {i}{28}\right ) \operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{\left (-\frac {1}{5}+\frac {7 i}{5}\right ) \left (t -5\right )}+\frac {\left (-7 \cos \left (\frac {7 t}{5}\right )-\sin \left (\frac {7 t}{5}\right )\right ) {\mathrm e}^{-\frac {t}{5}}}{14}-\frac {\operatorname {Heaviside}\left (t -5\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 91

DSolve[{D[x[t],{t,2}]+4/10*D[x[t],t]+2*x[t]==1-UnitStep[t-5],{x[0]==0,Derivative[1][x][0 ]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to -\frac {1}{14} e^{-t/5} \left (-\theta (5-t) \left (7 e^{t/5}+e \sin \left (7-\frac {7 t}{5}\right )-7 e \cos \left (7-\frac {7 t}{5}\right )\right )+e \sin \left (7-\frac {7 t}{5}\right )+\sin \left (\frac {7 t}{5}\right )-7 e \cos \left (7-\frac {7 t}{5}\right )+7 \cos \left (\frac {7 t}{5}\right )\right ) \]