Internal problem ID [10227]
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 }\left (t \right )&=c y \left (t \right )-b z \left (t \right )\\ y^{\prime }\left (t \right )&=a z \left (t \right )-c x \left (t \right )\\ z^{\prime }\left (t \right )&=b x \left (t \right )-a y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (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)],singsol=all)
\begin{align*} x \left (t \right ) &= 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 \left (t \right ) &= -\frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a c -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} 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 ) c_{3} a^{2} b -c_{1} b^{3}-c_{1} b \,c^{2}}{a \left (b^{2}+c^{2}\right )} \\ z \left (t \right ) &= \frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a b -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} 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 ) c_{3} a^{2} c +c_{1} b^{2} c +c_{1} c^{3}}{a \left (b^{2}+c^{2}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.045 (sec). Leaf size: 1084
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 {e^{t \left (-\sqrt {-a^2-b^2-c^2}\right )} \left (2 a^2 c_1 e^{t \sqrt {-a^2-b^2-c^2}}+b^2 c_1 \left (e^{2 t \sqrt {-a^2-b^2-c^2}}+1\right )+c^2 c_1 \left (e^{2 t \sqrt {-a^2-b^2-c^2}}+1\right )-c \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right ) \left (c_2 \sqrt {-a^2-b^2-c^2} \left (e^{t \sqrt {-a^2-b^2-c^2}}+1\right )+a c_3 \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right )\right )-b \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right ) \left (a c_2 \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right )-c_3 \sqrt {-a^2-b^2-c^2} \left (e^{t \sqrt {-a^2-b^2-c^2}}+1\right )\right )\right )}{2 \left (a^2+b^2+c^2\right )} \\ y(t)\to \frac {e^{t \left (-\sqrt {-a^2-b^2-c^2}\right )} \left (a^2 c_2 \left (e^{2 t \sqrt {-a^2-b^2-c^2}}+1\right )-a \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right ) \left (b c_1 \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right )+c_3 \sqrt {-a^2-b^2-c^2} \left (e^{t \sqrt {-a^2-b^2-c^2}}+1\right )\right )+2 b^2 c_2 e^{t \sqrt {-a^2-b^2-c^2}}+c^2 c_2 \left (e^{2 t \sqrt {-a^2-b^2-c^2}}+1\right )+c \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right ) \left (c_1 \sqrt {-a^2-b^2-c^2} \left (e^{t \sqrt {-a^2-b^2-c^2}}+1\right )-b c_3 \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right )\right )\right )}{2 \left (a^2+b^2+c^2\right )} \\ z(t)\to \frac {e^{t \left (-\sqrt {-a^2-b^2-c^2}\right )} \left (a^2 c_3 \left (e^{2 t \sqrt {-a^2-b^2-c^2}}+1\right )-a \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right ) \left (c c_1 \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right )-c_2 \sqrt {-a^2-b^2-c^2} \left (e^{t \sqrt {-a^2-b^2-c^2}}+1\right )\right )-b \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right ) \left (c_1 \sqrt {-a^2-b^2-c^2} \left (e^{t \sqrt {-a^2-b^2-c^2}}+1\right )+c c_2 \left (e^{t \sqrt {-a^2-b^2-c^2}}-1\right )\right )+2 c^2 c_3 e^{t \sqrt {-a^2-b^2-c^2}}+b^2 c_3 \left (e^{2 t \sqrt {-a^2-b^2-c^2}}+1\right )\right )}{2 \left (a^2+b^2+c^2\right )} \\ \end{align*}