28.4.27 problem 7.27
Internal
problem
ID
[4559]
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
:
7.27
Date
solved
:
Sunday, March 30, 2025 at 03:26:15 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )-x \left (t \right )-2 y \left (t \right )&=0\\ x \left (t \right )-\frac {d}{d t}y \left (t \right )&=15 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = x_{0}\\ y \left (0\right ) = y_{0} \end{align*}
✓ Maple. Time used: 0.212 (sec). Leaf size: 165
ode:=[diff(x(t),t)-x(t)-2*y(t) = 0, x(t)-diff(y(t),t) = 15*cos(t)*Heaviside(t-Pi)];
ic:=x(0) = x__0y(0) = y__0;
dsolve([ode,ic]);
\begin{align*}
x \left (t \right ) &= \frac {x_{0} {\mathrm e}^{-t}}{3}+\frac {2 x_{0} {\mathrm e}^{2 t}}{3}+\frac {2 y_{0} {\mathrm e}^{2 t}}{3}-\frac {2 y_{0} {\mathrm e}^{-t}}{3}+3 \sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )+9 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )+4 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-2 \pi +2 t}+5 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{\pi -t} \\
y \left (t \right ) &= \frac {x_{0} {\mathrm e}^{2 t}}{3}-\frac {x_{0} {\mathrm e}^{-t}}{3}+\frac {2 y_{0} {\mathrm e}^{-t}}{3}+\frac {y_{0} {\mathrm e}^{2 t}}{3}-6 \sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )-3 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )+2 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-2 \pi +2 t}-5 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{\pi -t} \\
\end{align*}
✓ Mathematica. Time used: 0.357 (sec). Leaf size: 160
ode={D[x[t],t]-x[t]-2*y[t]==0,x[t]-D[y[t],t]==15*Cos[t]*UnitStep[t-Pi]};
ic={x[0]==x0,y[0]==y0};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{3} e^{-t} \left (3 \theta (t-\pi ) \left (4 e^{3 t-2 \pi }+3 e^t \sin (t)+9 e^t \cos (t)+5 e^{\pi }\right )+2 e^{3 t} \text {x0}+2 \left (e^{3 t}-1\right ) \text {y0}+\text {x0}\right ) \\
y(t)\to \frac {1}{3} e^{-t-2 \pi } \left (e^{2 \pi } \left (\left (e^{3 t}-1\right ) \text {x0}+\left (e^{3 t}+2\right ) \text {y0}\right )-3 \theta (t-\pi ) \left (-2 e^{3 t}+6 e^{t+2 \pi } \sin (t)+3 e^{t+2 \pi } \cos (t)+5 e^{3 \pi }\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 1.555 (sec). Leaf size: 144
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(x(t) - 15*cos(t)*Heaviside(t - pi) - Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - C_{1} e^{- t} + 2 C_{2} e^{2 t} + \frac {4 e^{2 t} \theta \left (t - \pi \right )}{e^{2 \pi }} + 3 \sin {\left (t \right )} \theta \left (t - \pi \right ) + 9 \cos {\left (t \right )} \theta \left (t - \pi \right ) + 5 e^{\pi } e^{- t} \theta \left (t - \pi \right ), \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{2 t} + \frac {2 e^{2 t} \theta \left (t - \pi \right )}{e^{2 \pi }} - 6 \sin {\left (t \right )} \theta \left (t - \pi \right ) - 3 \cos {\left (t \right )} \theta \left (t - \pi \right ) - 5 e^{\pi } e^{- t} \theta \left (t - \pi \right )\right ]
\]