28.4.5 problem 7.5
Internal
problem
ID
[4537]
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
:
7.5
Date
solved
:
Friday, March 14, 2025 at 01:27:31 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )-3 x \left (t \right )-4 y \left (t \right )&=0\\ x \left (t \right )+\frac {d^{2}}{d t^{2}}y \left (t \right )+y \left (t \right )&=0 \end{align*}
✓ Maple. Time used: 0.038 (sec). Leaf size: 69
ode:=[diff(diff(x(t),t),t)-3*x(t)-4*y(t) = 0, x(t)+diff(diff(y(t),t),t)+y(t) = 0];
dsolve(ode);
\begin{align*}
x &= {\mathrm e}^{t} c_{1} +c_{2} {\mathrm e}^{t} t +c_3 \,{\mathrm e}^{-t}+c_4 \,{\mathrm e}^{-t} t \\
y &= -\frac {{\mathrm e}^{t} c_{1}}{2}-\frac {c_{2} {\mathrm e}^{t} t}{2}-\frac {c_3 \,{\mathrm e}^{-t}}{2}-\frac {c_4 \,{\mathrm e}^{-t} t}{2}+\frac {c_{2} {\mathrm e}^{t}}{2}-\frac {{\mathrm e}^{-t} c_4}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.016 (sec). Leaf size: 169
ode={D[x[t],{t,2}]-3*x[t]-4*y[t]==0,x[t]+D[y[t],{t,2}]+y[t]==0};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (-t+e^{2 t} (t+1)+1\right )-2 c_4 \left (e^{2 t}-1\right )+t \left (c_2 \left (e^{2 t}+1\right )+2 c_3 \left (e^{2 t}-1\right )+2 c_4 \left (e^{2 t}+1\right )\right )\right ) \\
y(t)\to \frac {1}{4} e^{-t} \left (c_2 \left (e^{2 t}-1\right )+2 c_3 \left (e^{2 t}+1\right )+4 c_4 \left (e^{2 t}-1\right )-t \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )+2 c_3 \left (e^{2 t}-1\right )+2 c_4 \left (e^{2 t}+1\right )\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.179 (sec). Leaf size: 61
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-3*x(t) - 4*y(t) + Derivative(x(t), (t, 2)),0),Eq(x(t) + y(t) + Derivative(y(t), (t, 2)),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = 2 C_{1} t e^{t} - 2 C_{2} e^{- t} - 2 C_{3} t e^{- t} + 2 C_{4} e^{t}, \ y{\left (t \right )} = - C_{1} t e^{t} + C_{3} t e^{- t} + \left (C_{1} - C_{4}\right ) e^{t} + \left (C_{2} + C_{3}\right ) e^{- t}\right ]
\]