Internal problem ID [9510]
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*} a x^{\prime }\relax (t )&=\left (b -c \right ) y \relax (t ) z \relax (t )\\ b y^{\prime }\relax (t )&=\left (-a +c \right ) z \relax (t ) x \relax (t )\\ c z^{\prime }\relax (t )&=\left (a -b \right ) x \relax (t ) y \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.844 (sec). Leaf size: 1350
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)
\begin{align*} \{x \relax (t ) = 0\} \\ \{y \relax (t ) = 0\} \\ \{z \relax (t ) = c_{1}\} \\ \end{align*} \begin{align*} \{x \relax (t ) = 0\} \\ \{y \relax (t ) = c_{1}\} \\ \{z \relax (t ) = 0\} \\ \end{align*} \begin{align*} \{x \relax (t ) = c_{1}\} \\ \{y \relax (t ) = 0\} \\ \{z \relax (t ) = 0\} \\ \end{align*} \begin{align*} \left \{x \relax (t ) = \RootOf \left (-\left (\int _{}^{\textit {\_Z}}-\frac {2 b c \left (a^{2}-a b -c a +b c \right )}{\sqrt {b c \left (a^{2}-a b -c a +b c \right ) \left (-4 a^{4} \textit {\_a}^{4}+8 \textit {\_a}^{4} a^{3} b +8 \textit {\_a}^{4} a^{3} c -4 \textit {\_a}^{4} a^{2} b^{2}-16 \textit {\_a}^{4} a^{2} b c -4 \textit {\_a}^{4} a^{2} c^{2}+8 \textit {\_a}^{4} a \,b^{2} c +8 \textit {\_a}^{4} a b \,c^{2}-4 \textit {\_a}^{4} b^{2} c^{2}+16 \textit {\_a}^{2} a^{4} c_{2}-32 \textit {\_a}^{2} a^{3} b c_{2}-32 \textit {\_a}^{2} a^{3} c c_{2}+16 \textit {\_a}^{2} a^{2} b^{2} c_{2}+64 \textit {\_a}^{2} a^{2} b c c_{2}+16 \textit {\_a}^{2} a^{2} c^{2} c_{2}-32 \textit {\_a}^{2} a \,b^{2} c c_{2}-32 \textit {\_a}^{2} a b \,c^{2} c_{2}+16 \textit {\_a}^{2} b^{2} c^{2} c_{2}-16 a^{4} c_{2}^{2}+32 a^{3} b c_{2}^{2}+32 a^{3} c c_{2}^{2}-16 a^{2} b^{2} c_{2}^{2}-64 a^{2} b c c_{2}^{2}-16 a^{2} c^{2} c_{2}^{2}+32 a \,b^{2} c c_{2}^{2}+32 a b \,c^{2} c_{2}^{2}-16 b^{2} c^{2} c_{2}^{2}+b c c_{1}\right )}}d \textit {\_a} \right )+t +c_{3}\right ), x \relax (t ) = \RootOf \left (-\left (\int _{}^{\textit {\_Z}}\frac {2 b c \left (a^{2}-a b -c a +b c \right )}{\sqrt {b c \left (a^{2}-a b -c a +b c \right ) \left (-4 a^{4} \textit {\_a}^{4}+8 \textit {\_a}^{4} a^{3} b +8 \textit {\_a}^{4} a^{3} c -4 \textit {\_a}^{4} a^{2} b^{2}-16 \textit {\_a}^{4} a^{2} b c -4 \textit {\_a}^{4} a^{2} c^{2}+8 \textit {\_a}^{4} a \,b^{2} c +8 \textit {\_a}^{4} a b \,c^{2}-4 \textit {\_a}^{4} b^{2} c^{2}+16 \textit {\_a}^{2} a^{4} c_{2}-32 \textit {\_a}^{2} a^{3} b c_{2}-32 \textit {\_a}^{2} a^{3} c c_{2}+16 \textit {\_a}^{2} a^{2} b^{2} c_{2}+64 \textit {\_a}^{2} a^{2} b c c_{2}+16 \textit {\_a}^{2} a^{2} c^{2} c_{2}-32 \textit {\_a}^{2} a \,b^{2} c c_{2}-32 \textit {\_a}^{2} a b \,c^{2} c_{2}+16 \textit {\_a}^{2} b^{2} c^{2} c_{2}-16 a^{4} c_{2}^{2}+32 a^{3} b c_{2}^{2}+32 a^{3} c c_{2}^{2}-16 a^{2} b^{2} c_{2}^{2}-64 a^{2} b c c_{2}^{2}-16 a^{2} c^{2} c_{2}^{2}+32 a \,b^{2} c c_{2}^{2}+32 a b \,c^{2} c_{2}^{2}-16 b^{2} c^{2} c_{2}^{2}+b c c_{1}\right )}}d \textit {\_a} \right )+t +c_{3}\right )\right \} \\ \left \{y \relax (t ) = -\frac {\sqrt {-2 x \relax (t ) b \left (a b -c a -b^{2}+b c \right ) \left (-\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) b c +\sqrt {4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a^{2} b c -4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a \,b^{2} c -4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a b \,c^{2}+4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} b^{2} c^{2}+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right )^{2} b^{2} c^{2}}\right ) a}}{2 x \relax (t ) b \left (a b -c a -b^{2}+b c \right )}, y \relax (t ) = \frac {\sqrt {-2 x \relax (t ) b \left (a b -c a -b^{2}+b c \right ) \left (-\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) b c +\sqrt {4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a^{2} b c -4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a \,b^{2} c -4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a b \,c^{2}+4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} b^{2} c^{2}+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right )^{2} b^{2} c^{2}}\right ) a}}{2 x \relax (t ) b \left (a b -c a -b^{2}+b c \right )}, y \relax (t ) = -\frac {\sqrt {2}\, \sqrt {x \relax (t ) b \left (a b -c a -b^{2}+b c \right ) \left (\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) b c +\sqrt {4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a^{2} b c -4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a \,b^{2} c -4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a b \,c^{2}+4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} b^{2} c^{2}+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right )^{2} b^{2} c^{2}}\right ) a}}{2 x \relax (t ) b \left (a b -c a -b^{2}+b c \right )}, y \relax (t ) = \frac {\sqrt {2}\, \sqrt {x \relax (t ) b \left (a b -c a -b^{2}+b c \right ) \left (\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) b c +\sqrt {4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a^{2} b c -4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a \,b^{2} c -4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} a b \,c^{2}+4 \left (\frac {d}{d t}x \relax (t )\right )^{2} x \relax (t )^{2} b^{2} c^{2}+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right )^{2} b^{2} c^{2}}\right ) a}}{2 x \relax (t ) b \left (a b -c a -b^{2}+b c \right )}\right \} \\ \left \{z \relax (t ) = \frac {a \left (\frac {d}{d t}x \relax (t )\right )}{b y \relax (t )-c y \relax (t )}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.641 (sec). Leaf size: 1285
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} \sqrt {b c_1 (b-c)} \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right )}{\sqrt {a (a-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 {b (b-a) c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to \frac {\sqrt {2} \sqrt {c c_2 (b-c) \text {dn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2}}{\sqrt {c} \sqrt {b-c}} \\ x(t)\to \frac {\sqrt {2} \sqrt {b c_1 (b-c)} \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right )}{\sqrt {a (a-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 {b (b-a) c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to -\frac {\sqrt {2} \sqrt {c c_2 (b-c) \text {dn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (c_3-t)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2}}{\sqrt {c} \sqrt {b-c}} \\ x(t)\to \frac {\sqrt {2} \sqrt {b c_1 (b-c)} \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right )}{\sqrt {a (a-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 {b (b-a) c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to -\frac {\sqrt {2} \sqrt {c c_2 (b-c) \text {dn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2}}{\sqrt {c} \sqrt {b-c}} \\ x(t)\to \frac {\sqrt {2} \sqrt {b c_1 (b-c)} \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right )}{\sqrt {a (a-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 {b (b-a) c_1}{(a-c) c c_2}\right ){}^2\right )}}{\sqrt {b (b-c)}} \\ z(t)\to \frac {\sqrt {2} \sqrt {c c_2 (b-c) \text {dn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} (t-c_3)}{\sqrt {a} \sqrt {b}}|\frac {b (b-a) c_1}{(a-c) c c_2}\right ){}^2}}{\sqrt {c} \sqrt {b-c}} \\ \end{align*}