60.9.51 problem 1906
Internal
problem
ID
[11905]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1906
Date
solved
:
Monday, January 27, 2025 at 11:44:22 PM
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*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 127
dsolve([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)],singsol=all)
\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 &= \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*}
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 177
DSolve[{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]},{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*}