67.4.19 problem Problem 3(e)
Internal
problem
ID
[13984]
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(e)
Date
solved
:
Wednesday, March 05, 2025 at 10:24:41 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=5 \cos \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \end{align*}
Using Laplace method With initial conditions
\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-1 \end{align*}
✓ Maple. Time used: 11.174 (sec). Leaf size: 58
ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+2*y(t) = 5*cos(t)*(Heaviside(t)-Heaviside(t-1/2*Pi));
ic:=y(0) = 1, D(y)(0) = -1;
dsolve([ode,ic],y(t),method='laplace');
\[
y = -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{\frac {\pi }{2}-t}+\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (-\cos \left (t \right )-2 \sin \left (t \right )\right )-3 \,{\mathrm e}^{-t} \sin \left (t \right )+\cos \left (t \right )+2 \sin \left (t \right )
\]
✓ Mathematica. Time used: 0.052 (sec). Leaf size: 72
ode=D[y[t],{t,2}]+2*D[y[t],t]+2*y[t]==5*Cos[t]*(UnitStep[t]-UnitStep[t-Pi/2]);
ic={y[0]==1,Derivative[1][y][0] ==-1};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} \cos (t) & t<0 \\ e^{-t} \left (\left (-3+2 e^{\pi /2}\right ) \sin (t)-e^{\pi /2} \cos (t)\right ) & 2 t>\pi \\ \cos (t)+\left (2-3 e^{-t}\right ) \sin (t) & \text {True} \\ \end {array} \\ \end {array}
\]
✓ Sympy. Time used: 3.666 (sec). Leaf size: 99
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-5*(Heaviside(t) - Heaviside(t - pi/2))*cos(t) + 2*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0)
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -1}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (\left (- 3 \theta \left (t\right ) + 2 e^{\frac {\pi }{2}} \theta \left (t - \frac {\pi }{2}\right )\right ) \sin {\left (t \right )} + \left (- \theta \left (t\right ) - e^{\frac {\pi }{2}} \theta \left (t - \frac {\pi }{2}\right ) + 1\right ) \cos {\left (t \right )}\right ) e^{- t} + 2 \sin {\left (t \right )} \theta \left (t\right ) - 2 \sin {\left (t \right )} \theta \left (t - \frac {\pi }{2}\right ) + \cos {\left (t \right )} \theta \left (t\right ) - \cos {\left (t \right )} \theta \left (t - \frac {\pi }{2}\right )
\]