63.17.7 problem 10

Internal problem ID [13206]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.4 Impulsive sources. Exercises page 173
Problem number : 10
Date solved : Tuesday, January 28, 2025 at 05:12:24 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x&=\frac {\left (t -5\right ) \operatorname {Heaviside}\left (t -5\right )}{5}+\left (2-\frac {t}{5}\right ) \operatorname {Heaviside}\left (t -10\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: 10.347 (sec). Leaf size: 49

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

\[ x \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -5\right ) \sin \left (-10+2 t \right )}{40}+\frac {\operatorname {Heaviside}\left (t -10\right ) \sin \left (2 t -20\right )}{40}+\frac {\left (-2 t +20\right ) \operatorname {Heaviside}\left (t -10\right )}{40}+\frac {\left (t -5\right ) \operatorname {Heaviside}\left (t -5\right )}{20} \]

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 55

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

\[ x(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{40} (2 (t-5)+\sin (10-2 t)) & 5<t\leq 10 \\ \frac {1}{40} (\sin (10-2 t)-\sin (20-2 t)+10) & t>10 \\ \end {array} \\ \end {array} \]