14.18.2 problem 2
Internal
problem
ID
[2686]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
2.
Second
order
differential
equations.
Section
2.11,
Differential
equations
with
discontinuous
right-hand
sides.
Excercises
page
243
Problem
number
:
2
Date
solved
:
Tuesday, March 04, 2025 at 02:34:06 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+y^{\prime }+y&=\operatorname {Heaviside}\left (t -\pi \right )-\operatorname {Heaviside}\left (t -2 \pi \right ) \end{align*}
Using Laplace method With initial conditions
\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}
✓ Maple. Time used: 3.489 (sec). Leaf size: 167
ode:=diff(diff(y(t),t),t)+diff(y(t),t)+y(t) = Heaviside(t-Pi)-Heaviside(t-2*Pi);
ic:=y(0) = 1, D(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {\operatorname {Heaviside}\left (t -2 \pi \right ) \left (i \sqrt {3}+3\right ) {\mathrm e}^{-\frac {\left (1+i \sqrt {3}\right ) \left (t -2 \pi \right )}{2}}}{6}+\frac {\left (-3-i \sqrt {3}\right ) \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{\frac {\left (1+i \sqrt {3}\right ) \left (-t +\pi \right )}{2}}}{6}+\frac {\left (3-i \sqrt {3}\right ) \operatorname {Heaviside}\left (t -2 \pi \right ) {\mathrm e}^{\frac {\left (i \sqrt {3}-1\right ) \left (t -2 \pi \right )}{2}}}{6}+\frac {\operatorname {Heaviside}\left (t -\pi \right ) \left (i \sqrt {3}-3\right ) {\mathrm e}^{-\frac {\left (i \sqrt {3}-1\right ) \left (-t +\pi \right )}{2}}}{6}+\frac {\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{3}+{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )-\operatorname {Heaviside}\left (t -2 \pi \right )+\operatorname {Heaviside}\left (t -\pi \right )
\]
✓ Mathematica. Time used: 0.248 (sec). Leaf size: 308
ode=D[y[t],{t,2}]+D[y[t],t]+y[t]==UnitStep[t-Pi]-UnitStep[t-2*Pi];
ic={y[0]==1,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{3} e^{-t/2} \left (3 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) & t\leq \pi \\ \frac {1}{3} e^{-t/2} \left (-3 e^{\pi /2} \cos \left (\frac {1}{2} \sqrt {3} (\pi -t)\right )+3 e^{t/2}+3 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} e^{\pi /2} \sin \left (\frac {1}{2} \sqrt {3} (\pi -t)\right )+\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) & \pi <t\leq 2 \pi \\ \frac {1}{3} e^{-t/2} \left (-3 e^{\pi /2} \cos \left (\frac {1}{2} \sqrt {3} (\pi -t)\right )+3 e^{\pi } \cos \left (\frac {1}{2} \sqrt {3} (2 \pi -t)\right )+3 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} e^{\pi /2} \sin \left (\frac {1}{2} \sqrt {3} (\pi -t)\right )-\sqrt {3} e^{\pi } \sin \left (\frac {1}{2} \sqrt {3} (2 \pi -t)\right )+\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) & \text {True} \\ \end {array} \\ \end {array}
\]
✓ Sympy. Time used: 6.291 (sec). Leaf size: 133
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(y(t) + Heaviside(t - 2*pi) - Heaviside(t - pi) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0)
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (\frac {\sqrt {3} \sin {\left (\frac {\sqrt {3} t}{2} \right )}}{3} + \frac {2 \sqrt {3} e^{\pi } \sin {\left (\frac {\sqrt {3} t}{2} - \sqrt {3} \pi + \frac {\pi }{3} \right )} \theta \left (t - 2 \pi \right )}{3} - \frac {2 \sqrt {3} e^{\frac {\pi }{2}} \sin {\left (\frac {\sqrt {3} t}{2} - \frac {\sqrt {3} \pi }{2} + \frac {\pi }{3} \right )} \theta \left (t - \pi \right )}{3} + \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} - \theta \left (t - 2 \pi \right ) + \theta \left (t - \pi \right )
\]