76.20.8 problem 8
Internal
problem
ID
[17685]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
5.
The
Laplace
transform.
Section
5.6
(Differential
equations
with
Discontinuous
Forcing
Functions).
Problems
at
page
342
Problem
number
:
8
Date
solved
:
Monday, March 31, 2025 at 04:24:31 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}&=t -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (t -\frac {\pi }{2}\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: 0.345 (sec). Leaf size: 63
ode:=diff(diff(y(t),t),t)+diff(y(t),t)+5/4*y(t) = t-Heaviside(t-1/2*Pi)*(t-1/2*Pi);
ic:=y(0) = 0, D(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = -\frac {16}{25}-\frac {12 \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )+\frac {4 \sin \left (t \right )}{3}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}}{25}+\frac {2 \left (8-10 t +5 \pi \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )}{25}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}} \left (4 \cos \left (t \right )-3 \sin \left (t \right )\right )}{25}+\frac {4 t}{5}
\]
✓ Mathematica. Time used: 0.06 (sec). Leaf size: 241
ode=D[y[t],{t,2}]+D[y[t],t]+12/100*y[t]==t-UnitStep[t-Pi/2]*(t-Pi/2);
ic={y[0]==0,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {25}{234} e^{-\frac {1}{20} \left (5+\sqrt {13}\right ) (2 t+\pi )} \left (\left (325-95 \sqrt {13}\right ) e^{\frac {1}{20} \left (5+\sqrt {13}\right ) \pi }+5 \left (-65+19 \sqrt {13}\right ) e^{\frac {1}{10} \left (5+\sqrt {13}\right ) \pi }-5 \left (65+19 \sqrt {13}\right ) e^{\frac {\sqrt {13} t}{5}+\frac {\pi }{2}}+5 \left (65+19 \sqrt {13}\right ) e^{\frac {1}{20} \left (4 \sqrt {13} t+\left (5+\sqrt {13}\right ) \pi \right )}+39 e^{\frac {1}{20} \left (5+\sqrt {13}\right ) (2 t+\pi )} \pi \right ) & 2 t>\pi \\ \frac {25}{234} e^{-\frac {1}{10} \left (5+\sqrt {13}\right ) t} \left (26 e^{\frac {1}{10} \left (5+\sqrt {13}\right ) t} (3 t-25)+5 \left (65+19 \sqrt {13}\right ) e^{\frac {\sqrt {13} t}{5}}-95 \sqrt {13}+325\right ) & \text {True} \\ \end {array} \\ \end {array}
\]
✓ Sympy. Time used: 4.336 (sec). Leaf size: 104
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-t + (t - pi/2)*Heaviside(t - pi/2) + 5*y(t)/4 + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0)
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = - \frac {4 t \theta \left (t - \frac {\pi }{2}\right )}{5} + \frac {4 t}{5} + \left (\left (- \frac {16 e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right )}{25} - \frac {12}{25}\right ) \sin {\left (t \right )} + \left (- \frac {12 e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right )}{25} + \frac {16}{25}\right ) \cos {\left (t \right )}\right ) e^{- \frac {t}{2}} + \frac {16 \theta \left (t - \frac {\pi }{2}\right )}{25} + \frac {2 \pi \theta \left (t - \frac {\pi }{2}\right )}{5} - \frac {16}{25}
\]