10.16.8 problem 8
Internal
problem
ID
[1408]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Chapter
7.6,
Complex
Eigenvalues.
page
417
Problem
number
:
8
Date
solved
:
Tuesday, March 04, 2025 at 12:35:21 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.036 (sec). Leaf size: 145
ode:=[diff(x__1(t),t) = -3*x__1(t)+2*x__3(t), diff(x__2(t),t) = x__1(t)-x__2(t), diff(x__3(t),t) = -2*x__1(t)-x__2(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-2 t}+c_2 \,{\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )+c_3 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right ) \\
x_{2} \left (t \right ) &= -c_1 \,{\mathrm e}^{-2 t}-\frac {c_2 \,{\mathrm e}^{-t} \sqrt {2}\, \cos \left (\sqrt {2}\, t \right )}{2}+\frac {c_3 \,{\mathrm e}^{-t} \sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{2} \\
x_{3} \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{-2 t}}{2}+c_2 \,{\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )+\frac {c_2 \,{\mathrm e}^{-t} \sqrt {2}\, \cos \left (\sqrt {2}\, t \right )}{2}+c_3 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )-\frac {c_3 \,{\mathrm e}^{-t} \sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.033 (sec). Leaf size: 235
ode={D[ x1[t],t]==-3*x1[t]+0*x2[t]+2*x3[t],D[ x2[t],t]==1*x1[t]-1*x2[t]-0*x3[t],D[ x3[t],t]==-2*x1[t]-1*x2[t]+0*x3[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to \frac {1}{3} e^{-2 t} \left ((c_1+2 (c_2+c_3)) e^t \cos \left (\sqrt {2} t\right )-\sqrt {2} (2 c_1+c_2-2 c_3) e^t \sin \left (\sqrt {2} t\right )+2 (c_1-c_2-c_3)\right ) \\
\text {x2}(t)\to \frac {1}{6} e^{-2 t} \left (2 (2 c_1+c_2-2 c_3) e^t \cos \left (\sqrt {2} t\right )+\sqrt {2} (c_1+2 (c_2+c_3)) e^t \sin \left (\sqrt {2} t\right )+4 (-c_1+c_2+c_3)\right ) \\
\text {x3}(t)\to \frac {1}{6} e^{-2 t} \left (-2 (c_1-c_2-4 c_3) e^t \cos \left (\sqrt {2} t\right )-\sqrt {2} (5 c_1+4 c_2-2 c_3) e^t \sin \left (\sqrt {2} t\right )+2 (c_1-c_2-c_3)\right ) \\
\end{align*}
✓ Sympy. Time used: 0.293 (sec). Leaf size: 155
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
x__3 = Function("x__3")
ode=[Eq(3*x__1(t) - 2*x__3(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + x__2(t) + Derivative(x__2(t), t),0),Eq(2*x__1(t) + x__2(t) + Derivative(x__3(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = 2 C_{3} e^{- 2 t} + \left (\frac {2 C_{1}}{3} + \frac {\sqrt {2} C_{2}}{3}\right ) e^{- t} \cos {\left (\sqrt {2} t \right )} + \left (\frac {\sqrt {2} C_{1}}{3} - \frac {2 C_{2}}{3}\right ) e^{- t} \sin {\left (\sqrt {2} t \right )}, \ x^{2}{\left (t \right )} = - 2 C_{3} e^{- 2 t} - \left (\frac {C_{1}}{3} - \frac {\sqrt {2} C_{2}}{3}\right ) e^{- t} \cos {\left (\sqrt {2} t \right )} + \left (\frac {\sqrt {2} C_{1}}{3} + \frac {C_{2}}{3}\right ) e^{- t} \sin {\left (\sqrt {2} t \right )}, \ x^{3}{\left (t \right )} = C_{1} e^{- t} \cos {\left (\sqrt {2} t \right )} - C_{2} e^{- t} \sin {\left (\sqrt {2} t \right )} + C_{3} e^{- 2 t}\right ]
\]