Internal problem ID [9483]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1904.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=c y \relax (t )-b z \relax (t )\\ y^{\prime }\relax (t )&=a z \relax (t )-c x \relax (t )\\ z^{\prime }\relax (t )&=b x \relax (t )-a y \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.102 (sec). Leaf size: 312
dsolve({diff(x(t),t)=c*y(t)-b*z(t),diff(y(t),t)=a*z(t)-c*x(t),diff(z(t),t)=b*x(t)-a*y(t)},{x(t), y(t), z(t)}, singsol=all)
\[ x \relax (t ) = c_{1}+c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \] \[ y \relax (t ) = -\frac {\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} a c +\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} b -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} a c +\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a^{2} b -c_{1} b^{3}-c_{1} b \,c^{2}}{a \left (b^{2}+c^{2}\right )} \] \[ z \relax (t ) = \frac {\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} a b -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} a b -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a^{2} c +c_{1} b^{2} c +c_{1} c^{3}}{a \left (b^{2}+c^{2}\right )} \]
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 405
DSolve[{x'[t]==c*y[t]-b*z[t],y'[t]==a*z[t]-c*x[t],z'[t]==b*x[t]-a*y[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {\left (-a b c_2+c (c c_1-a c_3)+b^2 c_1\right ) \cosh \left (t \sqrt {-a^2-b^2-c^2}\right )+\sqrt {-a^2-b^2-c^2} (b c_3-c c_2) \sinh \left (t \sqrt {-a^2-b^2-c^2}\right )+a (a c_1+b c_2+c c_3)}{a^2+b^2+c^2} \\ y(t)\to \frac {\left (a^2 c_2-a b c_1+c (c c_2-b c_3)\right ) \cosh \left (t \sqrt {-a^2-b^2-c^2}\right )+\sqrt {-a^2-b^2-c^2} (c c_1-a c_3) \sinh \left (t \sqrt {-a^2-b^2-c^2}\right )+b (a c_1+b c_2+c c_3)}{a^2+b^2+c^2} \\ z(t)\to \frac {\left (c_3 \left (a^2+b^2\right )-c (a c_1+b c_2)\right ) \cosh \left (t \sqrt {-a^2-b^2-c^2}\right )+\sqrt {-a^2-b^2-c^2} (a c_2-b c_1) \sinh \left (t \sqrt {-a^2-b^2-c^2}\right )+c (a c_1+b c_2+c c_3)}{a^2+b^2+c^2} \\ \end{align*}