67.4.23 problem Problem 3(i)
Internal
problem
ID
[13988]
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(i)
Date
solved
:
Wednesday, March 05, 2025 at 10:24:48 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} 4 y^{\prime \prime }+4 y^{\prime }+5 y&=25 t \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 )&=2\\ y^{\prime }\left (0\right )&=2 \end{align*}
✓ Maple. Time used: 13.316 (sec). Leaf size: 86
ode:=4*diff(diff(y(t),t),t)+4*diff(y(t),t)+5*y(t) = 25*t*(Heaviside(t)-Heaviside(t-1/2*Pi));
ic:=y(0) = 2, D(y)(0) = 2;
dsolve([ode,ic],y(t),method='laplace');
\[
y = -4+\left (\frac {5}{4}-\frac {5 i}{8}\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \pi \,{\mathrm e}^{\left (\frac {1}{4}-\frac {i}{2}\right ) \left (-2 t +\pi \right )}+\left (\frac {5}{4}+\frac {5 i}{8}\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \pi \,{\mathrm e}^{\left (\frac {1}{4}+\frac {i}{2}\right ) \left (-2 t +\pi \right )}-3 \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )+\frac {4 \sin \left (t \right )}{3}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}+\left (4-5 t \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )+6 \cos \left (t \right ) {\mathrm e}^{-\frac {t}{2}}+5 t
\]
✓ Mathematica. Time used: 0.049 (sec). Leaf size: 101
ode=4*D[y[t],{t,2}]+4*D[y[t],t]+5*y[t]==25*t*(UnitStep[t]-UnitStep[t-Pi/2]);
ic={y[0]==2,Derivative[1][y][0] ==2};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 5 t+6 e^{-t/2} \cos (t)-4 & t\geq 0\land 2 t\leq \pi \\ e^{-t/2} (2 \cos (t)+3 \sin (t)) & t<0 \\ \frac {1}{4} e^{-t/2} \left (\left (24-e^{\pi /4} (12+5 \pi )\right ) \cos (t)+2 e^{\pi /4} (-8+5 \pi ) \sin (t)\right ) & \text {True} \\ \end {array} \\ \end {array}
\]
✓ Sympy. Time used: 3.879 (sec). Leaf size: 136
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-25*t*(Heaviside(t) - Heaviside(t - pi/2)) + 5*y(t) + 4*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)),0)
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 2}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = 5 t \theta \left (t\right ) - 5 t \theta \left (t - \frac {\pi }{2}\right ) + \left (\left (- 3 \theta \left (t\right ) - 4 e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right ) + \frac {5 \pi e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right )}{2} + 3\right ) \sin {\left (t \right )} + \left (4 \theta \left (t\right ) - \frac {5 \pi e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right )}{4} - 3 e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right ) + 2\right ) \cos {\left (t \right )}\right ) e^{- \frac {t}{2}} - 4 \theta \left (t\right ) + 4 \theta \left (t - \frac {\pi }{2}\right )
\]