Internal problem ID [12012]
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]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y=\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )+\operatorname {Heaviside}\left (-2+t \right )-\operatorname {Heaviside}\left (-3+t \right )} \] 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: 0.115 (sec). Leaf size: 199
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]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{3}-e^{-t} & 0\leq t\leq 1 \\ -\frac {1}{6} e^{-3 t} \left (1+3 e^{2 t}\right ) & t<0 \\ \frac {1}{6} e^{-3 t} \left (-e^3-6 e^{2 t}+3 e^{2 t+1}\right ) & 1<t\leq 2 \\ \frac {1}{6} e^{-3 t} \left (-e^3+e^6-6 e^{2 t}+2 e^{3 t}+3 e^{2 t+1}-3 e^{2 t+2}\right ) & 2<t\leq 3 \\ \frac {1}{6} e^{-3 t} \left (-e^3+e^6-e^9-6 e^{2 t}+3 e^{2 t+1}-3 e^{2 t+2}+3 e^{2 t+3}\right ) & \text {True} \\ \end {array} \\ \end {array} \]