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
:
Wednesday, March 05, 2025 at 05:35:45 AM
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: 0.798 (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,ic],y(t),method='laplace');
\[
y = \frac {1}{3}-\frac {\operatorname {Heaviside}\left (t -4\right )}{3}+\frac {{\mathrm e}^{-3 t}}{6}-\frac {{\mathrm e}^{-t}}{2}+\frac {\operatorname {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6}-\frac {\operatorname {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2}+\frac {\operatorname {Heaviside}\left (t -6\right )}{3}-\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}-\frac {\operatorname {Heaviside}\left (t -2\right )}{3}
\]
✓ Mathematica. Time used: 0.063 (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]
\[
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{6} e^{-3 t} \left (-1+e^t\right )^2 \left (1+2 e^t\right ) & t\leq 2 \\ -\frac {1}{6} e^{-3 t} \left (-1+e^2\right ) \left (1+e^2+e^4-3 e^{2 t}\right ) & 2<t\leq 4 \\ \frac {1}{6} e^{-3 t} \left (-1+e^2\right )^2 \left (1+e^2\right ) \left (1+e^2+2 e^4+e^6+2 e^8+e^{10}+e^{12}-3 e^{2 t}\right ) & t>6 \\ -\frac {1}{6} e^{-3 t} \left (-1+e^6+e^{12}+3 e^{2 t}+2 e^{3 t}-3 e^{2 t+2}-3 e^{2 t+4}\right ) & \text {True} \\ \end {array} \\ \end {array}
\]
✓ Sympy. Time used: 1.195 (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)
\[
y{\left (t \right )} = \left (- \frac {e^{6} \theta \left (t - 6\right )}{2} + \frac {e^{4} \theta \left (t - 4\right )}{2} + \frac {e^{2} \theta \left (t - 2\right )}{2} - \frac {1}{2}\right ) e^{- t} + \left (\frac {e^{18} \theta \left (t - 6\right )}{6} - \frac {e^{12} \theta \left (t - 4\right )}{6} - \frac {e^{6} \theta \left (t - 2\right )}{6} + \frac {1}{6}\right ) e^{- 3 t} + \frac {\theta \left (t - 6\right )}{3} - \frac {\theta \left (t - 4\right )}{3} - \frac {\theta \left (t - 2\right )}{3} + \frac {1}{3}
\]