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
:
Monday, January 27, 2025 at 09:23:28 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )-x \left (t \right )-2 y&=0\\ x \left (t \right )-y^{\prime }&=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*}
✓ Solution by Maple
Time used: 0.059 (sec). Leaf size: 165
dsolve([diff(x(t),t)-x(t)-2*y(t) = 0, x(t)-diff(y(t),t) = 15*cos(t)*Heaviside(t-Pi), x(0) = x__0, y(0) = y__0], singsol=all)
\begin{align*}
x &= \frac {x_{0} {\mathrm e}^{-t}}{3}+\frac {2 \,{\mathrm e}^{2 t} x_{0}}{3}+\frac {2 y_{0} {\mathrm e}^{2 t}}{3}-\frac {2 y_{0} {\mathrm e}^{-t}}{3}+9 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )+3 \sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )+4 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 t -2 \pi }+5 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-t +\pi } \\
y &= -\frac {x_{0} {\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t} x_{0}}{3}+\frac {2 y_{0} {\mathrm e}^{-t}}{3}+\frac {y_{0} {\mathrm e}^{2 t}}{3}-3 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )-6 \sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )+2 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 t -2 \pi }-5 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-t +\pi } \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.357 (sec). Leaf size: 160
DSolve[{D[x[t],t]-x[t]-2*y[t]==0,x[t]-D[y[t],t]==15*Cos[t]*UnitStep[t-Pi]},{x[0]==x0,y[0]==y0},{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*}