75.27.6 problem 781

Internal problem ID [17163]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 20. The method of elimination. Exercises page 212
Problem number : 781
Date solved : Thursday, March 13, 2025 at 09:18:08 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.076 (sec). Leaf size: 55

ode:=[diff(x(t),t) = -y(t)+z(t), diff(y(t),t) = z(t), diff(z(t),t) = -x(t)+z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -c_{2} \cos \left (t \right )+\sin \left (t \right ) c_{3} +c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{3} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}-c_{2} \cos \left (t \right )+\sin \left (t \right ) c_{3} \\ z &= c_{1} {\mathrm e}^{t}+c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{3} \\ \end{align*}

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 112

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

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

✓ Sympy. Time used: 0.110 (sec). Leaf size: 49

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

\[ \left [ x{\left (t \right )} = - \left (C_{1} - C_{2}\right ) \sin {\left (t \right )} + \left (C_{1} + C_{2}\right ) \cos {\left (t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} + C_{2} \sin {\left (t \right )} + C_{3} e^{t}, \ z{\left (t \right )} = - C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} + C_{3} e^{t}\right ] \]