52.6.20 problem 70

Internal problem ID [8355]
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
Date solved : Monday, January 27, 2025 at 03:49:26 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 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 ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.798 (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)
 

\[ y = \frac {1}{3}-\frac {\operatorname {Heaviside}\left (t -4\right )}{3}+\frac {{\mathrm e}^{-3 t}}{6}-\frac {{\mathrm e}^{-t}}{2}+\frac {\operatorname {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6}-\frac {\operatorname {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2}+\frac {\operatorname {Heaviside}\left (t -6\right )}{3}-\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}-\frac {\operatorname {Heaviside}\left (t -2\right )}{3} \]

✓ Solution by Mathematica

Time used: 0.063 (sec). Leaf size: 175

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+3*y[t]==1-UnitStep[t-2]-UnitStep[t-4]+UnitStep[t-6],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{6} e^{-3 t} \left (-1+e^t\right )^2 \left (1+2 e^t\right ) & t\leq 2 \\ -\frac {1}{6} e^{-3 t} \left (-1+e^2\right ) \left (1+e^2+e^4-3 e^{2 t}\right ) & 2<t\leq 4 \\ \frac {1}{6} e^{-3 t} \left (-1+e^2\right )^2 \left (1+e^2\right ) \left (1+e^2+2 e^4+e^6+2 e^8+e^{10}+e^{12}-3 e^{2 t}\right ) & t>6 \\ -\frac {1}{6} e^{-3 t} \left (-1+e^6+e^{12}+3 e^{2 t}+2 e^{3 t}-3 e^{2 t+2}-3 e^{2 t+4}\right ) & \text {True} \\ \end {array} \\ \end {array} \]