60.9.56 problem 1911
Internal
problem
ID
[11910]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1911
Date
solved
:
Tuesday, January 28, 2025 at 06:24:05 PM
CAS
classification
:
system_of_ODEs
\begin{align*} a t \left (\frac {d}{d t}x \left (t \right )\right )&=b c \left (y \left (t \right )-z \left (t \right )\right )\\ b t \left (\frac {d}{d t}y \left (t \right )\right )&=c a \left (z \left (t \right )-x \left (t \right )\right )\\ c t \left (\frac {d}{d t}z \left (t \right )\right )&=a b \left (x \left (t \right )-y \left (t \right )\right ) \end{align*}
✓ Solution by Maple
Time used: 0.115 (sec). Leaf size: 321
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))],singsol=all)
\begin{align*}
x \left (t \right ) &= c_{1} +c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right )+c_3 \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) \\
y \left (t \right ) &= -\frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_3 a c -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{2} a c +\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{2} a^{2} b +\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_3 \,a^{2} b -c_{1} b^{3}-c_{1} b \,c^{2}}{b \left (b^{2}+c^{2}\right )} \\
z &= \frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_3 a b -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{2} a b -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{2} a^{2} c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_3 \,a^{2} c +c_{1} b^{2} c +c_{1} c^{3}}{\left (b^{2}+c^{2}\right ) c} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 715
DSolve[{a*t*D[x[t],t]==b*c*(y[t]-z[t]),b*t*D[y[t],t]==c*a*(z[t]-x[t]),c*t*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 {t^{-i \sqrt {a^2+b^2+c^2}} \left (a b^2 \left (c_1 \left (1+t^{2 i \sqrt {a^2+b^2+c^2}}\right )-c_2 \left (-1+t^{i \sqrt {a^2+b^2+c^2}}\right )^2\right )-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 c^2 \left (c_1 \left (1+t^{2 i \sqrt {a^2+b^2+c^2}}\right )-c_3 \left (-1+t^{i \sqrt {a^2+b^2+c^2}}\right )^2\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_1 \left (-1+t^{i \sqrt {a^2+b^2+c^2}}\right )^2-c_2 \left (1+t^{2 i \sqrt {a^2+b^2+c^2}}\right )\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 c^2 \left (c_2 \left (1+t^{2 i \sqrt {a^2+b^2+c^2}}\right )-c_3 \left (-1+t^{i \sqrt {a^2+b^2+c^2}}\right )^2\right )+2 b^3 c_2 t^{i \sqrt {a^2+b^2+c^2}}\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 (-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 )-a^2 c \left (c_1 \left (-1+t^{i \sqrt {a^2+b^2+c^2}}\right )^2-c_3 \left (1+t^{2 i \sqrt {a^2+b^2+c^2}}\right )\right )+b^2 c \left (c_3 \left (1+t^{2 i \sqrt {a^2+b^2+c^2}}\right )-c_2 \left (-1+t^{i \sqrt {a^2+b^2+c^2}}\right )^2\right )+2 c^3 c_3 t^{i \sqrt {a^2+b^2+c^2}}\right )}{2 c \left (a^2+b^2+c^2\right )} \\
\end{align*}