Internal problem ID [10984]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin.
CRC Press 2015
Section: Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number: Problem 3(j).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y-\operatorname {Heaviside}\left (t \right )+\operatorname {Heaviside}\left (-1+t \right )-\operatorname {Heaviside}\left (t -2\right )+\operatorname {Heaviside}\left (t -3\right )=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {2}{3}}, y^{\prime }\left (0\right ) = 1\right ] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 117
dsolve([diff(y(t),t$2)+4*diff(y(t),t)+3*y(t)=Heaviside(t)-Heaviside(t-1)+Heaviside(t-2)-Heaviside(t-3),y(0) = -2/3, D(y)(0) = 1],y(t), singsol=all)
\[ y \left (t \right ) = \frac {\left (-\frac {1}{3}-{\mathrm e}^{2+2 t} \operatorname {Heaviside}\left (t -2\right )+{\mathrm e}^{3+2 t} \operatorname {Heaviside}\left (t -3\right )+{\mathrm e}^{2 t +1} \operatorname {Heaviside}\left (t -1\right )+\frac {2 \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )+\operatorname {Heaviside}\left (t -2\right )-\operatorname {Heaviside}\left (t -3\right )\right ) {\mathrm e}^{3 t}}{3}-\frac {\operatorname {Heaviside}\left (t -3\right ) {\mathrm e}^{9}}{3}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{6}}{3}-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{3}}{3}+\left (-\operatorname {Heaviside}\left (t \right )-1\right ) {\mathrm e}^{2 t}+\frac {\operatorname {Heaviside}\left (t \right )}{3}\right ) {\mathrm e}^{-3 t}}{2} \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 153
DSolve[{y''[t]+4*y'[t]+3*y[t]==UnitStep[t]-UnitStep[t-1]+UnitStep[t-2]-UnitStep[t-3],{y[0]==-2/3,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{6} e^{-3 t} \left (2 e^{3 t} (\theta (1-t,t)+\theta (3-t))-3 e^{2 t} \left (2 \theta (1-t,t)+e^3 \theta (3-t)-\theta (t)+e \left (-1+e-e^2\right )+3\right )-\left (\left (2 e^t+e^2\right ) \left (e^2-e^t\right )^2 \theta (2-t)\right )+\left (e^3-3 (e-2) e^{2 t}\right ) \theta (1-t)+e^9 \theta (3-t)+\theta (t)-e^9+e^6-e^3-1\right ) \\ \end{align*}