64.19.2 problem 2
Internal
problem
ID
[13572]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
7,
Systems
of
linear
differential
equations.
Section
7.7.
Exercises
page
375
Problem
number
:
2
Date
solved
:
Monday, March 31, 2025 at 08:01:32 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-y \left (t \right )-z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+3 y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-3 x \left (t \right )-6 y \left (t \right )+6 z \left (t \right ) \end{align*}
✓ Maple. Time used: 0.120 (sec). Leaf size: 73
ode:=[diff(x(t),t) = x(t)-y(t)-z(t), diff(y(t),t) = x(t)+3*y(t)+z(t), diff(z(t),t) = -3*x(t)-6*y(t)+6*z(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{2 t}+c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{5 t} \\
y \left (t \right ) &= -\frac {7 c_1 \,{\mathrm e}^{2 t}}{10}-c_2 \,{\mathrm e}^{3 t}-c_3 \,{\mathrm e}^{5 t} \\
z \left (t \right ) &= -\frac {3 c_1 \,{\mathrm e}^{2 t}}{10}-c_2 \,{\mathrm e}^{3 t}-3 c_3 \,{\mathrm e}^{5 t} \\
\end{align*}
✓ Mathematica. Time used: 0.051 (sec). Leaf size: 217
ode={D[x[t],t]==x[t]-y[t]-z[t],D[y[t],t]==x[t]+3*y[t]+z[t],D[z[t],t]==3*x[t]-6*y[t]+6*z[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to -\frac {1}{45} e^{2 t} \left (5 (c_1+10 c_2) e^{2 t} \cos \left (\sqrt {5} t\right )+\sqrt {5} (7 c_1-11 c_2+9 c_3) e^{2 t} \sin \left (\sqrt {5} t\right )-50 (c_1+c_2)\right ) \\
y(t)\to \frac {1}{45} e^{2 t} \left (5 (c_1+10 c_2) e^{2 t} \cos \left (\sqrt {5} t\right )+\sqrt {5} (7 c_1-11 c_2+9 c_3) e^{2 t} \sin \left (\sqrt {5} t\right )-5 (c_1+c_2)\right ) \\
z(t)\to (c_1+c_2) \left (-e^{2 t}\right )+(c_1+c_2+c_3) e^{4 t} \cos \left (\sqrt {5} t\right )+\frac {(c_1-8 c_2+2 c_3) e^{4 t} \sin \left (\sqrt {5} t\right )}{\sqrt {5}} \\
\end{align*}
✓ Sympy. Time used: 0.156 (sec). Leaf size: 78
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-x(t) + y(t) + z(t) + Derivative(x(t), t),0),Eq(-x(t) - 3*y(t) - z(t) + Derivative(y(t), t),0),Eq(3*x(t) + 6*y(t) - 6*z(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {10 C_{1} e^{2 t}}{3} - C_{2} e^{3 t} - \frac {C_{3} e^{5 t}}{3}, \ y{\left (t \right )} = \frac {7 C_{1} e^{2 t}}{3} + C_{2} e^{3 t} + \frac {C_{3} e^{5 t}}{3}, \ z{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{3 t} + C_{3} e^{5 t}\right ]
\]