67.4.24 problem Problem 3(j)

Internal problem ID [13997]
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 : Monday, March 31, 2025 at 08:21:16 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*}

✓ Maple. Time used: 0.261 (sec). Leaf size: 102

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

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

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

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

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

✓ Sympy. Time used: 1.277 (sec). Leaf size: 134

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

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