Internal problem ID [10226]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1903.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=\frac {b c y \left (t \right )}{a}-\frac {b c z \left (t \right )}{a}\\ y^{\prime }\left (t \right )&=-\frac {c a x \left (t \right )}{b}+\frac {c a z \left (t \right )}{b}\\ z^{\prime }\left (t \right )&=\frac {a b x \left (t \right )}{c}-\frac {a b y \left (t \right )}{c} \end {align*}
✓ Solution by Maple
Time used: 0.266 (sec). Leaf size: 312
dsolve([a*diff(x(t),t)=b*c*(y(t)-z(t)),b*diff(y(t),t)=c*a*(z(t)-x(t)),c*diff(z(t),t)=a*b*(x(t)-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}}{b \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}}{\left (b^{2}+c^{2}\right ) c} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.084 (sec). Leaf size: 736
DSolve[{a*x'[t]==b*c*(y[t]-z[t]),b*y'[t]==c*a*(z[t]-x[t]),c*z'[t]==a*b*(x[t]-y[t])},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {e^{-i t \sqrt {a^2+b^2+c^2}} \left (a b^2 \left (c_1 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_2 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2\right )+a c^2 \left (c_1 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_3 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2\right )-i b c (c_2-c_3) \sqrt {a^2+b^2+c^2} \left (-1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )+2 a^3 c_1 e^{i t \sqrt {a^2+b^2+c^2}}\right )}{2 a \left (a^2+b^2+c^2\right )} \\ y(t)\to \frac {e^{-i t \sqrt {a^2+b^2+c^2}} \left (-a^2 b \left (c_1 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2-c_2 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )\right )+b c^2 \left (c_2 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_3 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2\right )+i a c (c_1-c_3) \sqrt {a^2+b^2+c^2} \left (-1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )+2 b^3 c_2 e^{i t \sqrt {a^2+b^2+c^2}}\right )}{2 b \left (a^2+b^2+c^2\right )} \\ z(t)\to \frac {e^{-i t \sqrt {a^2+b^2+c^2}} \left (-a^2 c \left (c_1 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2-c_3 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )\right )+b^2 c \left (c_3 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_2 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2\right )-i a b (c_1-c_2) \sqrt {a^2+b^2+c^2} \left (-1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )+2 c^3 c_3 e^{i t \sqrt {a^2+b^2+c^2}}\right )}{2 c \left (a^2+b^2+c^2\right )} \\ \end{align*}