46.6.20 problem 70

Internal problem ID [9641]
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 : Tuesday, September 30, 2025 at 06:21:45 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*}

✓ Maple. Time used: 0.206 (sec). Leaf size: 108

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+3*y(t) = 1-Heaviside(t-2)-Heaviside(t-4)+Heaviside(t-6); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 175

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

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

✓ Sympy. Time used: 0.733 (sec). Leaf size: 117

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(3*y(t) - Heaviside(t - 6) + Heaviside(t - 4) + Heaviside(t - 2) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 1,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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