Internal problem ID [9478]
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 z \left (t \right )}{b}-\frac {c a x \left (t \right )}{b}\\ z^{\prime }\left (t \right )&=-\frac {a b y \left (t \right )}{c}+\frac {a b x \left (t \right )}{c} \end {align*}
✓ Solution by Maple
Time used: 0.172 (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))],[x(t), y(t), z(t)], singsol=all)
\[ x \left (t \right ) = -\frac {\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} b c +\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a \,c^{2}-\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} b c +\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a \,c^{2}-c_{1} a^{3}-c_{1} a \,b^{2}}{a \left (a^{2}+b^{2}\right )} \] \[ y \left (t \right ) = \frac {\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} a c -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} b \,c^{2}-\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} a c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} b \,c^{2}+a^{2} b c_{1} +c_{1} b^{3}}{b \left (a^{2}+b^{2}\right )} \] \[ z \left (t \right ) = c_{1} +c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \]
✓ Solution by Mathematica
Time used: 0.052 (sec). Leaf size: 374
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 {a \left (b^2 (c_1-c_2)+c^2 (c_1-c_3)\right ) \cos \left (t \sqrt {a^2+b^2+c^2}\right )+b c (c_2-c_3) \sqrt {a^2+b^2+c^2} \sin \left (t \sqrt {a^2+b^2+c^2}\right )+a \left (a^2 c_1+b^2 c_2+c^2 c_3\right )}{a \left (a^2+b^2+c^2\right )} \\ y(t)\to \frac {b \left (a^2 (c_2-c_1)+c^2 (c_2-c_3)\right ) \cos \left (t \sqrt {a^2+b^2+c^2}\right )+a c (c_3-c_1) \sqrt {a^2+b^2+c^2} \sin \left (t \sqrt {a^2+b^2+c^2}\right )+b \left (a^2 c_1+b^2 c_2+c^2 c_3\right )}{b \left (a^2+b^2+c^2\right )} \\ z(t)\to \frac {c \left (a^2 (c_3-c_1)+b^2 (c_3-c_2)\right ) \cos \left (t \sqrt {a^2+b^2+c^2}\right )+a b (c_1-c_2) \sqrt {a^2+b^2+c^2} \sin \left (t \sqrt {a^2+b^2+c^2}\right )+c \left (a^2 c_1+b^2 c_2+c^2 c_3\right )}{c \left (a^2+b^2+c^2\right )} \\ \end{align*}