Internal problem ID [10253]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1931.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-\frac {y \left (t \right ) z \left (t \right ) c}{a}+\frac {y \left (t \right ) z \left (t \right ) b}{a}\\ y^{\prime }\left (t \right )&=-\frac {z \left (t \right ) x \left (t \right ) a}{b}+\frac {z \left (t \right ) x \left (t \right ) c}{b}\\ z^{\prime }\left (t \right )&=\frac {x \left (t \right ) y \left (t \right ) a}{c}-\frac {x \left (t \right ) y \left (t \right ) b}{c} \end {align*}
✓ Solution by Maple
Time used: 0.86 (sec). Leaf size: 1356
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*} \\ \\ \\ \text {Expression too large to display} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.217 (sec). Leaf size: 1461
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 {\sqrt {2} b c_1 \sqrt {a (a-c)} (c-b) \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{a (c-a) \sqrt {b c_1 (b-c)}} \\ y(t)\to -\frac {\sqrt {2} \sqrt {-b c_1 (b-c) \left (-1+\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to \frac {\sqrt {2} \sqrt {\frac {(b-c) \left (b c_1 (b-a) \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2+c c_2 (c-a)\right )}{c-a}}}{\sqrt {c} \sqrt {b-c}} \\ x(t)\to \frac {\sqrt {2} b c_1 \sqrt {a (a-c)} (c-b) \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{a (c-a) \sqrt {b c_1 (b-c)}} \\ y(t)\to \frac {\sqrt {2} \sqrt {-b c_1 (b-c) \left (-1+\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to -\frac {\sqrt {2} \sqrt {\frac {(b-c) \left (b c_1 (b-a) \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2+c c_2 (c-a)\right )}{c-a}}}{\sqrt {c} \sqrt {b-c}} \\ x(t)\to \frac {\sqrt {2} b c_1 \sqrt {a (a-c)} (c-b) \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{a (c-a) \sqrt {b c_1 (b-c)}} \\ y(t)\to -\frac {\sqrt {2} \sqrt {-b c_1 (b-c) \left (-1+\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to -\frac {\sqrt {2} \sqrt {\frac {(b-c) \left (b c_1 (b-a) \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2+c c_2 (c-a)\right )}{c-a}}}{\sqrt {c} \sqrt {b-c}} \\ x(t)\to \frac {\sqrt {2} b c_1 \sqrt {a (a-c)} (c-b) \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{a (c-a) \sqrt {b c_1 (b-c)}} \\ y(t)\to \frac {\sqrt {2} \sqrt {-b c_1 (b-c) \left (-1+\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to \frac {\sqrt {2} \sqrt {\frac {(b-c) \left (b c_1 (b-a) \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2+c c_2 (c-a)\right )}{c-a}}}{\sqrt {c} \sqrt {b-c}} \\ \end{align*}