60.9.49 problem 1904

Internal problem ID [11828]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1904
Date solved : Wednesday, March 05, 2025 at 03:07:32 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.108 (sec). Leaf size: 311

ode:=[diff(x(t),t) = c*y(t)-b*z(t), diff(y(t),t) = a*z(t)-c*x(t), diff(z(t),t) = b*x(t)-a*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}}{a \left (b^{2}+c^{2}\right )} \\ 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}}{a \left (b^{2}+c^{2}\right )} \\ \end{align*}

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 1084

ode={D[x[t],t]==c*y[t]-b*z[t],D[y[t],t]==a*z[t]-c*x[t],D[z[t],t]==b*x[t]-a*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.539 (sec). Leaf size: 303

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

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