60.9.48 problem 1903

Internal problem ID [11827]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1903
Date solved : Wednesday, March 05, 2025 at 03:07:30 PM
CAS classification : system_of_ODEs

\begin{align*} a \left (\frac {d}{d t}x \left (t \right )\right )&=b c \left (y \left (t \right )-z \left (t \right )\right )\\ b \left (\frac {d}{d t}y \left (t \right )\right )&=c a \left (z \left (t \right )-x \left (t \right )\right )\\ c \left (\frac {d}{d t}z \left (t \right )\right )&=a b \left (x \left (t \right )-y \left (t \right )\right ) \end{align*}

✓ Maple. Time used: 0.173 (sec). Leaf size: 311

ode:=[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))]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} +c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+c_3 \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) \\ y \left (t \right ) &= -\frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_3 a c -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a c +\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} b +\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_3 \,a^{2} b -c_{1} b^{3}-c_{1} b \,c^{2}}{\left (b^{2}+c^{2}\right ) b} \\ z &= \frac {\sqrt {a^{2}+b^{2}+c^{2}}\, \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_3 a b -\sqrt {a^{2}+b^{2}+c^{2}}\, \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a b -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_{2} a^{2} c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ) c_3 \,a^{2} c +c_{1} b^{2} c +c_{1} c^{3}}{\left (b^{2}+c^{2}\right ) c} \\ \end{align*}

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 736

ode={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])}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {e^{-i t \sqrt {a^2+b^2+c^2}} \left (a b^2 \left (c_1 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_2 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2\right )+a c^2 \left (c_1 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_3 \left (-1+e^{i t \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+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )+2 a^3 c_1 e^{i t \sqrt {a^2+b^2+c^2}}\right )}{2 a \left (a^2+b^2+c^2\right )} \\ y(t)\to \frac {e^{-i t \sqrt {a^2+b^2+c^2}} \left (-a^2 b \left (c_1 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2-c_2 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )\right )+b c^2 \left (c_2 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_3 \left (-1+e^{i t \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+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )+2 b^3 c_2 e^{i t \sqrt {a^2+b^2+c^2}}\right )}{2 b \left (a^2+b^2+c^2\right )} \\ z(t)\to \frac {e^{-i t \sqrt {a^2+b^2+c^2}} \left (-a^2 c \left (c_1 \left (-1+e^{i t \sqrt {a^2+b^2+c^2}}\right )^2-c_3 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )\right )+b^2 c \left (c_3 \left (1+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )-c_2 \left (-1+e^{i t \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+e^{2 i t \sqrt {a^2+b^2+c^2}}\right )+2 c^3 c_3 e^{i t \sqrt {a^2+b^2+c^2}}\right )}{2 c \left (a^2+b^2+c^2\right )} \\ \end{align*}

✓ Sympy. Time used: 0.621 (sec). Leaf size: 255

from sympy import * 
t = symbols("t") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(a*Derivative(x(t), t) - b*c*(y(t) - z(t)),0),Eq(-a*c*(-x(t) + z(t)) + b*Derivative(y(t), t),0),Eq(-a*b*(x(t) - y(t)) + c*Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} - \frac {C_{2} c \left (a c + b \sqrt {- a^{2} - b^{2} - c^{2}}\right ) e^{- t \sqrt {- a^{2} - b^{2} - c^{2}}}}{a \left (a^{2} + b^{2}\right )} - \frac {C_{3} c \left (a c - b \sqrt {- a^{2} - b^{2} - c^{2}}\right ) e^{t \sqrt {- a^{2} - b^{2} - c^{2}}}}{a \left (a^{2} + b^{2}\right )}, \ y{\left (t \right )} = C_{1} + \frac {C_{2} c \left (a \sqrt {- a^{2} - b^{2} - c^{2}} - b c\right ) e^{- t \sqrt {- a^{2} - b^{2} - c^{2}}}}{b \left (a^{2} + b^{2}\right )} - \frac {C_{3} c \left (a \sqrt {- a^{2} - b^{2} - c^{2}} + b c\right ) e^{t \sqrt {- a^{2} - b^{2} - c^{2}}}}{b \left (a^{2} + b^{2}\right )}, \ z{\left (t \right )} = C_{1} + C_{2} e^{- t \sqrt {- a^{2} - b^{2} - c^{2}}} + C_{3} e^{t \sqrt {- a^{2} - b^{2} - c^{2}}}\right ] \]