Internal problem ID [5934]
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]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+4 y^{\prime }+3 y-1+\theta \left (t -2\right )+\theta \left (t -4\right )-\theta \left (t -6\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 88
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)
\[ y \relax (t ) = -\frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{-3 t}}{6}+\frac {\left (-3 \,{\mathrm e}^{-t +6}+{\mathrm e}^{-3 t +18}+2\right ) \theta \left (t -6\right )}{6}+\frac {\left (3 \,{\mathrm e}^{4-t}-{\mathrm e}^{-3 t +12}-2\right ) \theta \left (t -4\right )}{6}+\frac {1}{3}+\frac {\left (-{\mathrm e}^{-3 t +6}+3 \,{\mathrm e}^{-t +2}-2\right ) \theta \left (t -2\right )}{6} \]
✓ Solution by Mathematica
Time used: 0.027 (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*}