Internal problem ID [5933]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS. Page
297
Problem number: 70.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y-1+\operatorname {Heaviside}\left (t -2\right )+\operatorname {Heaviside}\left (t -4\right )-\operatorname {Heaviside}\left (t -6\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 108
dsolve([diff(y(t),t$2)+4*diff(y(t),t)+3*y(t)=1-Heaviside(t-2)-Heaviside(t-4)+Heaviside(t-6),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = -\frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{-3 t}}{6}-\frac {\operatorname {Heaviside}\left (t -2\right )}{3}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}-\frac {\operatorname {Heaviside}\left (t -4\right )}{3}+\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}+\frac {\operatorname {Heaviside}\left (t -6\right )}{3}-\frac {\operatorname {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2}+\frac {1}{3}-\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}-\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\operatorname {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6} \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 129
DSolve[{y''[t]+4*y'[t]+3*y[t]==1-UnitStep[t-2]-UnitStep[t-4]+UnitStep[t-6],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{6} e^{-3 t} \left (\left (2 e^t+e^2\right ) \left (e^2-e^t\right )^2 \theta (2-t)+\left (e^4-e^t\right )^2 \left (2 e^t+e^4\right ) \theta (4-t)-\left (e^6-e^t\right )^2 \left (2 e^t+e^6\right ) \theta (6-t)-3 \left (e^2-1\right )^2 \left (1+e^2\right ) e^{2 t}+\left (e^6-1\right )^2 \left (1+e^6\right )\right ) \\ \end{align*}