28.4.28 problem 7.28
Internal
problem
ID
[4560]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
7.
Systems
of
linear
differential
equations.
Problems
at
page
351
Problem
number
:
7.28
Date
solved
:
Tuesday, March 04, 2025 at 06:52:24 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )-x \left (t \right )+y \left (t \right )&=2 \sin \left (t \right ) \left (1-\operatorname {Heaviside}\left (t -\pi \right )\right )\\ 2 x \left (t \right )-\frac {d}{d t}y \left (t \right )-y \left (t \right )&=0 \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 0 \end{align*}
✓ Maple. Time used: 0.053 (sec). Leaf size: 113
ode:=[diff(x(t),t)-x(t)+y(t) = 2*sin(t)*(1-Heaviside(t-Pi)), 2*x(t)-diff(y(t),t)-y(t) = 0];
ic:=x(0) = 0y(0) = 0;
dsolve([ode,ic]);
\begin{align*}
x &= -\cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \pi +\sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \pi +\cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) t -\sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) t -\cos \left (t \right ) t -\sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )+\sin \left (t \right ) t +\sin \left (t \right ) \\
y &= -2 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \pi +2 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) t -2 \cos \left (t \right ) t -2 \sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )+2 \sin \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.116 (sec). Leaf size: 56
ode={D[x[t],t]-x[t]+y[t]==2*Sin[t]*(1-UnitStep[t-Pi]),2*x[t]-D[y[t],t]-y[t]==0};
ic={x[0]==0,y[0]==0};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \pi (\sin (t)-\cos (t)) & t\geq \pi \\ (t+1) \sin (t)-t \cos (t) & \text {True} \\ \end {array} \\ \end {array} \\
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -2 \pi \cos (t) & t\geq \pi \\ 2 \sin (t)-2 t \cos (t) & \text {True} \\ \end {array} \\ \end {array} \\
\end{align*}
✓ Sympy. Time used: 1.466 (sec). Leaf size: 287
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq((2*Heaviside(t - pi) - 2)*sin(t) - x(t) + y(t) + Derivative(x(t), t),0),Eq(2*x(t) - y(t) - Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - t \sin {\left (t \right )} \theta \left (t - \pi \right ) + t \sin {\left (t \right )} + t \cos {\left (t \right )} \theta \left (t - \pi \right ) - t \cos {\left (t \right )} + \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) \cos {\left (t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) \sin {\left (t \right )} + \frac {\sin {\left (t \right )} \sin {\left (2 t \right )} \theta \left (t - \pi \right )}{2} - \frac {\sin {\left (t \right )} \sin {\left (2 t \right )}}{2} + \sin {\left (t \right )} \cos ^{2}{\left (t \right )} \theta \left (t - \pi \right ) - \sin {\left (t \right )} \cos ^{2}{\left (t \right )} - \left (1 - \pi \right ) \sin {\left (t \right )} \theta \left (t - \pi \right ) - \frac {\sin {\left (2 t \right )} \cos {\left (t \right )} \theta \left (t - \pi \right )}{2} + \frac {\sin {\left (2 t \right )} \cos {\left (t \right )}}{2} + \cos ^{3}{\left (t \right )} \theta \left (t - \pi \right ) - \cos ^{3}{\left (t \right )} - \left (1 + \pi \right ) \cos {\left (t \right )} \theta \left (t - \pi \right ), \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} + 2 t \cos {\left (t \right )} \theta \left (t - \pi \right ) - 2 t \cos {\left (t \right )} + 2 \sin {\left (t \right )} \cos ^{2}{\left (t \right )} \theta \left (t - \pi \right ) - 2 \sin {\left (t \right )} \cos ^{2}{\left (t \right )} - 2 \sin {\left (t \right )} \theta \left (t - \pi \right ) - \sin {\left (2 t \right )} \cos {\left (t \right )} \theta \left (t - \pi \right ) + \sin {\left (2 t \right )} \cos {\left (t \right )} - 2 \pi \cos {\left (t \right )} \theta \left (t - \pi \right )\right ]
\]