54.9.50 problem 1906
Internal
problem
ID
[13128]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1906
Date
solved
:
Wednesday, October 01, 2025 at 03:34:46 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 )&=y \left (t \right )+z \left (t \right )-x \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=z \left (t \right )+x \left (t \right )-y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.134 (sec). Leaf size: 127
ode:=[diff(x(t),t) = x(t)+y(t)-z(t), diff(y(t),t) = y(t)+z(t)-x(t), diff(z(t),t) = z(t)+x(t)-y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{t} \left (c_1 +\sin \left (\sqrt {3}\, t \right ) c_2 +\cos \left (\sqrt {3}\, t \right ) c_3 \right ) \\
y \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (\sin \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_3 -\cos \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_2 +\sin \left (\sqrt {3}\, t \right ) c_2 +\cos \left (\sqrt {3}\, t \right ) c_3 -2 c_1 \right )}{2} \\
z \left (t \right ) &= \frac {{\mathrm e}^{t} \left (\sin \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_3 -\cos \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_2 -\sin \left (\sqrt {3}\, t \right ) c_2 -\cos \left (\sqrt {3}\, t \right ) c_3 +2 c_1 \right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.018 (sec). Leaf size: 177
ode={D[x[t],t]==x[t]+y[t]-z[t],D[y[t],t]==y[t]+z[t]-x[t],D[z[t],t]==z[t]+x[t]-y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{3} e^t \left ((2 c_1-c_2-c_3) \cos \left (\sqrt {3} t\right )+\sqrt {3} (c_2-c_3) \sin \left (\sqrt {3} t\right )+c_1+c_2+c_3\right )\\ y(t)&\to \frac {1}{3} e^t \left (-(c_1-2 c_2+c_3) \cos \left (\sqrt {3} t\right )-\sqrt {3} (c_1-c_3) \sin \left (\sqrt {3} t\right )+c_1+c_2+c_3\right )\\ z(t)&\to \frac {1}{3} e^t \left (-(c_1+c_2-2 c_3) \cos \left (\sqrt {3} t\right )+\sqrt {3} (c_1-c_2) \sin \left (\sqrt {3} t\right )+c_1+c_2+c_3\right ) \end{align*}
✓ Sympy. Time used: 0.147 (sec). Leaf size: 143
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) - y(t) - z(t) + Derivative(y(t), t),0),Eq(-x(t) + y(t) - z(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} e^{t} - \left (\frac {C_{2}}{2} + \frac {\sqrt {3} C_{3}}{2}\right ) e^{t} \cos {\left (\sqrt {3} t \right )} - \left (\frac {\sqrt {3} C_{2}}{2} - \frac {C_{3}}{2}\right ) e^{t} \sin {\left (\sqrt {3} t \right )}, \ y{\left (t \right )} = C_{1} e^{t} - \left (\frac {C_{2}}{2} - \frac {\sqrt {3} C_{3}}{2}\right ) e^{t} \cos {\left (\sqrt {3} t \right )} + \left (\frac {\sqrt {3} C_{2}}{2} + \frac {C_{3}}{2}\right ) e^{t} \sin {\left (\sqrt {3} t \right )}, \ z{\left (t \right )} = C_{1} e^{t} + C_{2} e^{t} \cos {\left (\sqrt {3} t \right )} - C_{3} e^{t} \sin {\left (\sqrt {3} t \right )}\right ]
\]