66.14.12 problem 16
Internal
problem
ID
[16162]
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
:
16
Date
solved
:
Thursday, October 02, 2025 at 10:43:05 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-y\\ y^{\prime }&=-2 y+3 z \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )+3 y-z \left (t \right ) \end{align*}
✓ Maple. Time used: 0.112 (sec). Leaf size: 170
ode:=[diff(x(t),t) = 2*x(t)-y(t), diff(y(t),t) = -2*y(t)+3*z(t), diff(z(t),t) = -x(t)+3*y(t)-z(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= -c_2 \,{\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t}-c_3 \,{\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t}-\frac {2 c_2 \,{\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}+\frac {2 c_3 \,{\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}+c_1 \,{\mathrm e}^{t} \\
y \left (t \right ) &= c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t}+c_3 \,{\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t} \\
z \left (t \right ) &= \frac {2 c_2 \,{\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}-\frac {2 c_3 \,{\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}+c_1 \,{\mathrm e}^{t}+\frac {c_2 \,{\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t}}{3}+\frac {c_3 \,{\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t}}{3} \\
\end{align*}
✓ Mathematica. Time used: 0.013 (sec). Leaf size: 474
ode={D[x[t],t]==2*x[t]-1*y[t]+0*z[t],D[y[t],t]==0*x[t]-2*y[t]+3*z[t],D[z[t],t]==-1*x[t]+3*y[t]-1*z[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{16} e^{-\left (\left (1+2 \sqrt {3}\right ) t\right )} \left (c_1 \left (\left (5+3 \sqrt {3}\right ) e^{4 \sqrt {3} t}+6 e^{2 \left (1+\sqrt {3}\right ) t}+5-3 \sqrt {3}\right )-2 c_2 \left (\left (1+\sqrt {3}\right ) e^{4 \sqrt {3} t}-2 e^{2 \left (1+\sqrt {3}\right ) t}+1-\sqrt {3}\right )-c_3 \left (\left (3+\sqrt {3}\right ) e^{4 \sqrt {3} t}-6 e^{2 \left (1+\sqrt {3}\right ) t}+3-\sqrt {3}\right )\right )\\ y(t)&\to \frac {1}{16} e^{-\left (\left (1+2 \sqrt {3}\right ) t\right )} \left (c_1 \left (-\left (3+\sqrt {3}\right ) e^{4 \sqrt {3} t}+6 e^{2 \left (1+\sqrt {3}\right ) t}-3+\sqrt {3}\right )+2 c_2 \left (-\left (\sqrt {3}-3\right ) e^{4 \sqrt {3} t}+2 e^{2 \left (1+\sqrt {3}\right ) t}+3+\sqrt {3}\right )+3 c_3 \left (\left (\sqrt {3}-1\right ) e^{4 \sqrt {3} t}+2 e^{2 \left (1+\sqrt {3}\right ) t}-1-\sqrt {3}\right )\right )\\ z(t)&\to -\frac {1}{48} e^{-\left (\left (1+2 \sqrt {3}\right ) t\right )} \left (c_1 \left (\left (9+7 \sqrt {3}\right ) e^{4 \sqrt {3} t}-18 e^{2 \left (1+\sqrt {3}\right ) t}+9-7 \sqrt {3}\right )-2 c_2 \left (\left (5 \sqrt {3}-3\right ) e^{4 \sqrt {3} t}+6 e^{2 \left (1+\sqrt {3}\right ) t}-3-5 \sqrt {3}\right )+3 c_3 \left (\left (\sqrt {3}-5\right ) e^{4 \sqrt {3} t}-6 e^{2 \left (1+\sqrt {3}\right ) t}-5-\sqrt {3}\right )\right ) \end{align*}
✓ Sympy. Time used: 0.184 (sec). Leaf size: 146
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-2*x(t) + y(t) + Derivative(x(t), t),0),Eq(2*y(t) - 3*z(t) + Derivative(y(t), t),0),Eq(x(t) - 3*y(t) + z(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (4 \sqrt {3} + 9\right ) e^{- t \left (1 - 2 \sqrt {3}\right )}}{11} - \frac {C_{2} \left (9 - 4 \sqrt {3}\right ) e^{- t \left (1 + 2 \sqrt {3}\right )}}{11} + C_{3} e^{t}, \ y{\left (t \right )} = - \frac {3 C_{1} \left (1 - 2 \sqrt {3}\right ) e^{- t \left (1 - 2 \sqrt {3}\right )}}{11} - \frac {3 C_{2} \left (1 + 2 \sqrt {3}\right ) e^{- t \left (1 + 2 \sqrt {3}\right )}}{11} + C_{3} e^{t}, \ z{\left (t \right )} = C_{1} e^{- t \left (1 - 2 \sqrt {3}\right )} + C_{2} e^{- t \left (1 + 2 \sqrt {3}\right )} + C_{3} e^{t}\right ]
\]