Internal problem ID [9482]
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 }\relax (t )&=\frac {b c \left (y \relax (t )-z \relax (t )\right )}{a}\\ y^{\prime }\relax (t )&=-\frac {c a \left (x \relax (t )-z \relax (t )\right )}{b}\\ z^{\prime }\relax (t )&=\frac {a b \left (x \relax (t )-y \relax (t )\right )}{c} \end {align*}
✓ Solution by Maple
Time used: 0.142 (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)
\[ x \relax (t ) = c_{1}+c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \] \[ y \relax (t ) = -\frac {\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} a c +\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} b -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} a c +\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a^{2} b -c_{1} b^{3}-c_{1} b \,c^{2}}{\left (b^{2}+c^{2}\right ) b} \] \[ z \relax (t ) = \frac {\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} a b -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} a b -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{3} a^{2} c +c_{1} b^{2} c +c_{1} c^{3}}{\left (b^{2}+c^{2}\right ) c} \]
✓ Solution by Mathematica
Time used: 0.058 (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*}