67.4.24 problem Problem 3(j)

Internal problem ID [14068]
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)
Date solved : Tuesday, January 28, 2025 at 06:13:33 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+3 y&=\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )+\operatorname {Heaviside}\left (t -2\right )-\operatorname {Heaviside}\left (t -3\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-{\frac {2}{3}}\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 10.520 (sec). Leaf size: 102

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)
 

\[ y = \frac {1}{3}-\frac {\operatorname {Heaviside}\left (t -3\right )}{3}-{\mathrm e}^{-t}-\frac {\operatorname {Heaviside}\left (t -3\right ) {\mathrm e}^{-3 t +9}}{6}+\frac {\operatorname {Heaviside}\left (t -3\right ) {\mathrm e}^{-t +3}}{2}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}-\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}+\frac {\operatorname {Heaviside}\left (t -2\right )}{3}+\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t}}{2}-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{6}-\frac {\operatorname {Heaviside}\left (t -1\right )}{3} \]

✓ Solution by Mathematica

Time used: 0.067 (sec). Leaf size: 199

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+3*y[t]==UnitStep[t]-UnitStep[t-1]+UnitStep[t-2]-UnitStep[t-3],{y[0]==-2/3,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{3}-e^{-t} & 0\leq t\leq 1 \\ -\frac {1}{6} e^{-3 t} \left (1+3 e^{2 t}\right ) & t<0 \\ \frac {1}{6} e^{-3 t} \left (-e^3-6 e^{2 t}+3 e^{2 t+1}\right ) & 1<t\leq 2 \\ \frac {1}{6} e^{-3 t} \left (-e^3+e^6-6 e^{2 t}+2 e^{3 t}+3 e^{2 t+1}-3 e^{2 t+2}\right ) & 2<t\leq 3 \\ \frac {1}{6} e^{-3 t} \left (-e^3+e^6-e^9-6 e^{2 t}+3 e^{2 t+1}-3 e^{2 t+2}+3 e^{2 t+3}\right ) & \text {True} \\ \end {array} \\ \end {array} \]