75.29.7 problem 808

Internal problem ID [17182]
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 22. Integration of homogeneous linear systems with constant coefficients. Eulers method. Exercises page 230
Problem number : 808
Date solved : Thursday, March 13, 2025 at 09:18:22 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+2 y \left (t \right )-z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )-y \left (t \right )+2 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.165 (sec). Leaf size: 51

ode:=[diff(x(t),t) = 2*x(t)-y(t)+z(t), diff(y(t),t) = x(t)+2*y(t)-z(t), diff(z(t),t) = x(t)-y(t)+2*z(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 99

ode={D[x[t],t]==2*x[t]-y[t]+z[t],D[y[t],t]==x[t]+2*y[t]-z[t],D[z[t],t]==x[t]-y[t]+2*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.125 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-2*x(t) + y(t) - z(t) + Derivative(x(t), t),0),Eq(-x(t) - 2*y(t) + z(t) + Derivative(y(t), t),0),Eq(-x(t) + y(t) - 2*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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