Internal problem ID [11530]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+4 x=\frac {\left (t -5\right ) \operatorname {Heaviside}\left (t -5\right )}{5}+\left (2-\frac {t}{5}\right ) \operatorname {Heaviside}\left (-10+t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 5.594 (sec). Leaf size: 43
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)
\[ x \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -10\right ) \sin \left (2 t -20\right )}{40}-\frac {\operatorname {Heaviside}\left (t -5\right ) \sin \left (2 t -10\right )}{40}+\frac {\left (-2 t +20\right ) \operatorname {Heaviside}\left (t -10\right )}{40}+\frac {\left (t -5\right ) \operatorname {Heaviside}\left (t -5\right )}{20} \]
✓ Solution by Mathematica
Time used: 0.085 (sec). Leaf size: 55
DSolve[{x''[t]+4*x[t]==1/5*(t-5)*UnitStep[t-5]+(1-1/5*(t-5))*UnitStep[t-10],{x[0]==0,x'[0]==0}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{40} (2 (t-5)+\sin (10-2 t)) & 5<t\leq 10 \\ \frac {1}{40} (\sin (10-2 t)-\sin (20-2 t)+10) & t>10 \\ \end {array} \\ \end {array} \]