66.14.14 problem 18
Internal
problem
ID
[16164]
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
:
18
Date
solved
:
Thursday, October 02, 2025 at 10:43:07 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=-10 x \left (t \right )+10 y\\ y^{\prime }&=28 x \left (t \right )-y\\ z^{\prime }\left (t \right )&=-\frac {8 z \left (t \right )}{3} \end{align*}
✓ Maple. Time used: 0.123 (sec). Leaf size: 94
ode:=[diff(x(t),t) = -10*x(t)+10*y(t), diff(y(t),t) = 28*x(t)-y(t), diff(z(t),t) = -8/3*z(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (-11+\sqrt {1201}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (11+\sqrt {1201}\right ) t}{2}} \\
y \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{\frac {\left (-11+\sqrt {1201}\right ) t}{2}} \sqrt {1201}}{20}-\frac {c_2 \,{\mathrm e}^{-\frac {\left (11+\sqrt {1201}\right ) t}{2}} \sqrt {1201}}{20}+\frac {9 c_1 \,{\mathrm e}^{\frac {\left (-11+\sqrt {1201}\right ) t}{2}}}{20}+\frac {9 c_2 \,{\mathrm e}^{-\frac {\left (11+\sqrt {1201}\right ) t}{2}}}{20} \\
z \left (t \right ) &= c_3 \,{\mathrm e}^{-\frac {8 t}{3}} \\
\end{align*}
✓ Mathematica. Time used: 0.019 (sec). Leaf size: 312
ode={D[x[t],t]==-10*x[t]+10*y[t]+0*z[t],D[y[t],t]==28*x[t]-1*y[t]+0*z[t],D[z[t],t]==0*x[t]+0*y[t]-8/3*z[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {e^{-\frac {1}{2} \left (11+\sqrt {1201}\right ) t} \left (c_1 \left (\left (1201-9 \sqrt {1201}\right ) e^{\sqrt {1201} t}+1201+9 \sqrt {1201}\right )+20 \sqrt {1201} c_2 \left (e^{\sqrt {1201} t}-1\right )\right )}{2402}\\ y(t)&\to \frac {e^{-\frac {1}{2} \left (11+\sqrt {1201}\right ) t} \left (56 \sqrt {1201} c_1 \left (e^{\sqrt {1201} t}-1\right )+c_2 \left (\left (1201+9 \sqrt {1201}\right ) e^{\sqrt {1201} t}+1201-9 \sqrt {1201}\right )\right )}{2402}\\ z(t)&\to c_3 e^{-8 t/3}\\ x(t)&\to \frac {e^{-\frac {1}{2} \left (11+\sqrt {1201}\right ) t} \left (c_1 \left (\left (1201-9 \sqrt {1201}\right ) e^{\sqrt {1201} t}+1201+9 \sqrt {1201}\right )+20 \sqrt {1201} c_2 \left (e^{\sqrt {1201} t}-1\right )\right )}{2402}\\ y(t)&\to \frac {e^{-\frac {1}{2} \left (11+\sqrt {1201}\right ) t} \left (56 \sqrt {1201} c_1 \left (e^{\sqrt {1201} t}-1\right )+c_2 \left (\left (1201+9 \sqrt {1201}\right ) e^{\sqrt {1201} t}+1201-9 \sqrt {1201}\right )\right )}{2402}\\ z(t)&\to 0 \end{align*}
✓ Sympy. Time used: 0.147 (sec). Leaf size: 87
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(10*x(t) - 10*y(t) + Derivative(x(t), t),0),Eq(-28*x(t) + y(t) + Derivative(y(t), t),0),Eq(8*z(t)/3 + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (9 - \sqrt {1201}\right ) e^{- \frac {t \left (11 - \sqrt {1201}\right )}{2}}}{56} - \frac {C_{2} \left (9 + \sqrt {1201}\right ) e^{- \frac {t \left (11 + \sqrt {1201}\right )}{2}}}{56}, \ y{\left (t \right )} = C_{1} e^{- \frac {t \left (11 - \sqrt {1201}\right )}{2}} + C_{2} e^{- \frac {t \left (11 + \sqrt {1201}\right )}{2}}, \ z{\left (t \right )} = C_{3} e^{- \frac {8 t}{3}}\right ]
\]