Internal problem ID [9490]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1911.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=-\frac {b c \left (z \relax (t )-y \relax (t )\right )}{a t}\\ y^{\prime }\relax (t )&=\frac {c a \left (z \relax (t )-x \relax (t )\right )}{b t}\\ z^{\prime }\relax (t )&=\frac {a b \left (x \relax (t )-y \relax (t )\right )}{c t} \end {align*}
✓ Solution by Maple
Time used: 0.104 (sec). Leaf size: 322
dsolve({a*t*diff(x(t),t)=b*c*(y(t)-z(t)),b*t*diff(y(t),t)=c*a*(z(t)-x(t)),c*t*diff(z(t),t)=a*b*(x(t)-y(t))},{x(t), y(t), z(t)}, singsol=all)
\[ x \relax (t ) = c_{1}+c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \relax (t )\right )+c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \relax (t )\right ) \] \[ y \relax (t ) = \frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \relax (t )\right ) c_{2} a c -\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \relax (t )\right ) c_{3} a c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \relax (t )\right ) c_{3} a^{2} b -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \relax (t )\right ) c_{2} a^{2} b +c_{1} b^{3}+c_{1} b \,c^{2}}{b \left (b^{2}+c^{2}\right )} \] \[ z \relax (t ) = -\frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \relax (t )\right ) c_{2} a b -\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \relax (t )\right ) c_{3} a b +\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \relax (t )\right ) c_{3} a^{2} c +\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \relax (t )\right ) c_{2} a^{2} c -c_{1} b^{2} c -c_{1} c^{3}}{\left (b^{2}+c^{2}\right ) c} \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 692
DSolve[{a*t*x'[t]==b*c*(y[t]-z[t]),b*t*y'[t]==c*a*(z[t]-x[t]),c*t*z'[t]==a*b*(x[t]-y[t])},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {t^{-i \sqrt {a^2+b^2+c^2}} \left (-i b c (c_2-c_3) \sqrt {a^2+b^2+c^2} \left (-1+t^{2 i \sqrt {a^2+b^2+c^2}}\right )+a \left (2 \left (b^2 c_2+c^2 c_3\right ) t^{i \sqrt {a^2+b^2+c^2}}+\left (b^2 (c_1-c_2)+c^2 (c_1-c_3)\right ) t^{2 i \sqrt {a^2+b^2+c^2}}+b^2 c_1-b^2 c_2+c^2 c_1-c^2 c_3\right )+2 a^3 c_1 t^{i \sqrt {a^2+b^2+c^2}}\right )}{2 a \left (a^2+b^2+c^2\right )} \\ y(t)\to \frac {t^{-i \sqrt {a^2+b^2+c^2}} \left (a^2 b \left (c_2 \left (1+t^{2 i \sqrt {a^2+b^2+c^2}}\right )-c_1 \left (-1+t^{i \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+t^{2 i \sqrt {a^2+b^2+c^2}}\right )+b \left (2 \left (b^2 c_2+c^2 c_3\right ) t^{i \sqrt {a^2+b^2+c^2}}+c^2 (c_2-c_3) t^{2 i \sqrt {a^2+b^2+c^2}}+c^2 (c_2-c_3)\right )\right )}{2 b \left (a^2+b^2+c^2\right )} \\ z(t)\to \frac {t^{-i \sqrt {a^2+b^2+c^2}} \left (a^2 c \left (c_3 \left (1+t^{2 i \sqrt {a^2+b^2+c^2}}\right )-c_1 \left (-1+t^{i \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+t^{2 i \sqrt {a^2+b^2+c^2}}\right )+c \left (2 \left (b^2 c_2+c^2 c_3\right ) t^{i \sqrt {a^2+b^2+c^2}}-b^2 (c_2-c_3) t^{2 i \sqrt {a^2+b^2+c^2}}+b^2 (c_3-c_2)\right )\right )}{2 c \left (a^2+b^2+c^2\right )} \\ \end{align*}