Internal problem ID [9485]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1906.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=x \relax (t )+y \relax (t )-z \relax (t )\\ y^{\prime }\relax (t )&=y \relax (t )+z \relax (t )-x \relax (t )\\ z^{\prime }\relax (t )&=z \relax (t )+x \relax (t )-y \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.092 (sec). Leaf size: 128
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)},{x(t), y(t), z(t)}, singsol=all)
\[ x \relax (t ) = {\mathrm e}^{t} \left (\sin \left (\sqrt {3}\, t \right ) c_{2}+\cos \left (\sqrt {3}\, t \right ) c_{3}+c_{1}\right ) \] \[ y \relax (t ) = -\frac {{\mathrm e}^{t} \left (\sqrt {3}\, \sin \left (\sqrt {3}\, t \right ) c_{3}-\sqrt {3}\, \cos \left (\sqrt {3}\, t \right ) c_{2}+\sin \left (\sqrt {3}\, t \right ) c_{2}+\cos \left (\sqrt {3}\, t \right ) c_{3}-2 c_{1}\right )}{2} \] \[ z \relax (t ) = \frac {{\mathrm e}^{t} \left (\sqrt {3}\, \sin \left (\sqrt {3}\, t \right ) c_{3}-\sqrt {3}\, \cos \left (\sqrt {3}\, t \right ) c_{2}-\sin \left (\sqrt {3}\, t \right ) c_{2}-\cos \left (\sqrt {3}\, t \right ) c_{3}+2 c_{1}\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.029 (sec). Leaf size: 176
DSolve[{x'[t]==x[t]+y[t]-z[t],y'[t]==y[t]+z[t]-x[t],z'[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_3-c_1) \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*}