10.19.10 problem 10
Internal
problem
ID
[1451]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Chapter
9.1,
The
Phase
Plane:
Linear
Systems.
page
505
Problem
number
:
10
Date
solved
:
Saturday, March 29, 2025 at 10:55:33 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-5 x_{1} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.138 (sec). Leaf size: 83
ode:=[diff(x__1(t),t) = x__1(t)+2*x__2(t), diff(x__2(t),t) = -5*x__1(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= \frac {{\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {\sqrt {39}\, t}{2}\right ) \sqrt {39}\, c_2 -\cos \left (\frac {\sqrt {39}\, t}{2}\right ) \sqrt {39}\, c_1 -\sin \left (\frac {\sqrt {39}\, t}{2}\right ) c_1 -\cos \left (\frac {\sqrt {39}\, t}{2}\right ) c_2 \right )}{10} \\
x_{2} \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {\sqrt {39}\, t}{2}\right ) c_1 +\cos \left (\frac {\sqrt {39}\, t}{2}\right ) c_2 \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.005 (sec). Leaf size: 54
ode={D[ x1[t],t]==1*x1[t]+2*x2[t],D[ x2[t],t]==-5*x1[t]-1*x2[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to c_1 \cos (3 t)+\frac {1}{3} (c_1+2 c_2) \sin (3 t) \\
\text {x2}(t)\to c_2 \cos (3 t)-\frac {1}{3} (5 c_1+c_2) \sin (3 t) \\
\end{align*}
✓ Sympy. Time used: 0.185 (sec). Leaf size: 92
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(-x__1(t) - 2*x__2(t) + Derivative(x__1(t), t),0),Eq(5*x__1(t) + Derivative(x__2(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - \left (\frac {C_{1}}{10} - \frac {\sqrt {39} C_{2}}{10}\right ) e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {39} t}{2} \right )} + \left (\frac {\sqrt {39} C_{1}}{10} + \frac {C_{2}}{10}\right ) e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {39} t}{2} \right )}, \ x^{2}{\left (t \right )} = C_{1} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {39} t}{2} \right )} - C_{2} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {39} t}{2} \right )}\right ]
\]