60.10.18 problem 1931
Internal
problem
ID
[11929]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1931
Date
solved
:
Tuesday, January 28, 2025 at 06:24:09 PM
CAS
classification
:
system_of_ODEs
\begin{align*} a \left (\frac {d}{d t}x \left (t \right )\right )&=\left (b -c \right ) y \left (t \right ) z \left (t \right )\\ b \left (\frac {d}{d t}y \left (t \right )\right )&=\left (c -a \right ) z \left (t \right ) x \left (t \right )\\ c \left (\frac {d}{d t}z \left (t \right )\right )&=\left (a -b \right ) x \left (t \right ) y \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 2.731 (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)],singsol=all)
\begin{align*}
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\\
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\text {Expression too large to display} \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.506 (sec). Leaf size: 1461
DSolve[{a*D[x[t],t]==(b-c)*y[t]*z[t],b*D[y[t],t]==(c-a)*z[t]*x[t],c*D[z[t],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*}