72.14.13 problem 17
Internal
problem
ID
[14797]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
3.
Linear
Systems.
Exercises
section
3.8
page
371
Problem
number
:
17
Date
solved
:
Thursday, March 13, 2025 at 04:19:09 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-4 x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=5 x \left (t \right )-5 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.086 (sec). Leaf size: 100
ode:=[diff(x(t),t) = -4*x(t)+3*y(t), diff(y(t),t) = -y(t)+z(t), diff(z(t),t) = 5*x(t)-5*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{-t} c_{1} +\frac {6 \,{\mathrm e}^{-2 t} \sin \left (t \right ) c_{2}}{5}-\frac {3 \,{\mathrm e}^{-2 t} \cos \left (t \right ) c_{2}}{5}+\frac {6 c_{3} {\mathrm e}^{-2 t} \cos \left (t \right )}{5}+\frac {3 c_{3} {\mathrm e}^{-2 t} \sin \left (t \right )}{5} \\
y &= {\mathrm e}^{-t} c_{1} +{\mathrm e}^{-2 t} \sin \left (t \right ) c_{2} +c_{3} {\mathrm e}^{-2 t} \cos \left (t \right ) \\
z &= -{\mathrm e}^{-2 t} \left (c_{2} \sin \left (t \right )+\sin \left (t \right ) c_{3} -c_{2} \cos \left (t \right )+\cos \left (t \right ) c_{3} \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.016 (sec). Leaf size: 152
ode={D[x[t],t]==-4*x[t]+3*y[t]+0*z[t],D[y[t],t]==0*x[t]-1*y[t]+1*z[t],D[z[t],t]==5*x[t]-5*y[t]+0*z[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{2} e^{-2 t} \left ((5 c_1-3 c_2+3 c_3) e^t-3 (c_1-c_2+c_3) \cos (t)-3 (3 c_1-3 c_2+c_3) \sin (t)\right ) \\
y(t)\to \frac {1}{2} e^{-2 t} \left ((5 c_1-3 c_2+3 c_3) e^t+(-5 c_1+5 c_2-3 c_3) \cos (t)-(5 c_1-5 c_2+c_3) \sin (t)\right ) \\
z(t)\to e^{-2 t} (c_3 \cos (t)+(5 c_1-5 c_2+2 c_3) \sin (t)) \\
\end{align*}
✓ Sympy. Time used: 0.194 (sec). Leaf size: 104
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(4*x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(y(t) - z(t) + Derivative(y(t), t),0),Eq(-5*x(t) + 5*y(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{3} e^{- t} + \left (\frac {3 C_{1}}{10} + \frac {9 C_{2}}{10}\right ) e^{- 2 t} \sin {\left (t \right )} - \left (\frac {9 C_{1}}{10} - \frac {3 C_{2}}{10}\right ) e^{- 2 t} \cos {\left (t \right )}, \ y{\left (t \right )} = C_{3} e^{- t} - \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- 2 t} \cos {\left (t \right )} + \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- 2 t} \sin {\left (t \right )}, \ z{\left (t \right )} = C_{1} e^{- 2 t} \cos {\left (t \right )} - C_{2} e^{- 2 t} \sin {\left (t \right )}\right ]
\]