28.4.29 problem 7.29
Internal
problem
ID
[4561]
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
:
7.29
Date
solved
:
Tuesday, March 04, 2025 at 06:52:25 PM
CAS
classification
:
system_of_ODEs
\begin{align*} 2 \frac {d}{d t}x \left (t \right )+x \left (t \right )-5 \frac {d}{d t}y \left (t \right )-4 y \left (t \right )&=28 \,{\mathrm e}^{t} \operatorname {Heaviside}\left (t -2\right )\\ 3 \frac {d}{d t}x \left (t \right )-2 x \left (t \right )-4 \frac {d}{d t}y \left (t \right )+y \left (t \right )&=0 \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = 0 \end{align*}
✓ Maple. Time used: 0.020 (sec). Leaf size: 85
ode:=[2*diff(x(t),t)+x(t)-5*diff(y(t),t)-4*y(t) = 28*exp(t)*Heaviside(t-2), 3*diff(x(t),t)-2*x(t)-4*diff(y(t),t)+y(t) = 0];
ic:=x(0) = 2y(0) = 0;
dsolve([ode,ic]);
\begin{align*}
x &= -{\mathrm e}^{-t}+3 \,{\mathrm e}^{t}+5 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{4-t}+7 \,{\mathrm e}^{t} \operatorname {Heaviside}\left (t -2\right )-6 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{t} t \\
y &= {\mathrm e}^{t}-{\mathrm e}^{-t}+5 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{4-t}-{\mathrm e}^{t} \operatorname {Heaviside}\left (t -2\right )-2 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{t} t \\
\end{align*}
✓ Mathematica. Time used: 1.895 (sec). Leaf size: 302
ode={D[x[t],t]+x[t]-5*D[y[t],t]-4*y[t]==28*Exp[t]*UnitStep[t-2],3*D[x[t],t]-2*x[t]-4*D[y[t],t]+y[t]==0};
ic={x[0]==2,y[0]==0};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{309} e^{\frac {1}{22} \left (-\sqrt {309} t+t-2 \sqrt {309}\right )} \left (3 e^{\frac {\sqrt {309}}{11}} \left (\left (103+9 \sqrt {309}\right ) e^{\frac {\sqrt {309} t}{11}}+103-9 \sqrt {309}\right )-14 \left (-\left (309+13 \sqrt {309}\right ) e^{\frac {1}{11} \left (\sqrt {309} t+21\right )}+618 e^{\frac {1}{22} \left (\left (21+\sqrt {309}\right ) t+2 \sqrt {309}\right )}+\left (13 \sqrt {309}-309\right ) e^{\frac {1}{11} \left (21+2 \sqrt {309}\right )}\right ) \theta (t-2)\right ) \\
y(t)\to \frac {1}{309} e^{\frac {1}{22} \left (-\sqrt {309} t+t-2 \sqrt {309}\right )} \left (10 \sqrt {309} e^{\frac {\sqrt {309}}{11}} \left (e^{\frac {\sqrt {309} t}{11}}-1\right )-14 \left (-\left (103+\sqrt {309}\right ) e^{\frac {1}{11} \left (\sqrt {309} t+21\right )}+206 e^{\frac {1}{22} \left (\left (21+\sqrt {309}\right ) t+2 \sqrt {309}\right )}+\left (\sqrt {309}-103\right ) e^{\frac {1}{11} \left (21+2 \sqrt {309}\right )}\right ) \theta (t-2)\right ) \\
\end{align*}
✓ Sympy. Time used: 0.469 (sec). Leaf size: 114
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(x(t) - 4*y(t) - 28*exp(t)*Heaviside(t - 2) + 2*Derivative(x(t), t) - 5*Derivative(y(t), t),0),Eq(-2*x(t) + y(t) + 3*Derivative(x(t), t) - 4*Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} e^{- t} + 3 C_{2} e^{t} - 6 \left (\begin {cases} 0 & \text {for}\: \left |{t}\right | < 2 \\t {G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {\frac {t}{2}} \right )} & \text {otherwise} \end {cases}\right ) e^{t} - 5 e^{t} \theta \left (t - 2\right ) + 5 e^{4} e^{- t} \theta \left (t - 2\right ), \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t} - 2 \left (\begin {cases} 0 & \text {for}\: \left |{t}\right | < 2 \\t {G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {\frac {t}{2}} \right )} & \text {otherwise} \end {cases}\right ) e^{t} - 5 e^{t} \theta \left (t - 2\right ) + 5 e^{4} e^{- t} \theta \left (t - 2\right )\right ]
\]