87.21.18 problem 18

Internal problem ID [23732]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 178
Problem number : 18
Date solved : Thursday, October 02, 2025 at 09:44:41 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=3 x-y \left (t \right )\\ y^{\prime }\left (t \right )&=-x+2 y \left (t \right )-z \left (t \right )\\ z^{\prime }\left (t \right )&=-y \left (t \right )+3 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.066 (sec). Leaf size: 60

ode:=[diff(x(t),t) = 3*x(t)-y(t), diff(y(t),t) = -x(t)+2*y(t)-z(t), diff(z(t),t) = -y(t)+3*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -c_3 \,{\mathrm e}^{4 t}+{\mathrm e}^{3 t} c_1 +\frac {c_2 \,{\mathrm e}^{t}}{2} \\ y \left (t \right ) &= c_2 \,{\mathrm e}^{t}+c_3 \,{\mathrm e}^{4 t} \\ z \left (t \right ) &= -c_3 \,{\mathrm e}^{4 t}-{\mathrm e}^{3 t} c_1 +\frac {c_2 \,{\mathrm e}^{t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 160

ode={D[x[t],t]==3*x[t]-y[t],D[y[t],t]==-x[t]+2*y[t]-z[t],D[z[t],t]==-y[t]+3*z[t] }; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{6} e^t \left (c_3 \left (2 e^t+1\right ) \left (e^t-1\right )^2+c_1 \left (3 e^{2 t}+2 e^{3 t}+1\right )-2 c_2 \left (e^{3 t}-1\right )\right )\\ y(t)&\to -\frac {1}{3} e^t \left (c_1 \left (e^{3 t}-1\right )-c_2 \left (e^{3 t}+2\right )+c_3 \left (e^{3 t}-1\right )\right )\\ z(t)&\to \frac {1}{6} e^t \left (c_1 \left (2 e^t+1\right ) \left (e^t-1\right )^2-2 c_2 \left (e^{3 t}-1\right )+c_3 \left (3 e^{2 t}+2 e^{3 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-3*x(t) + y(t) + Derivative(x(t), t),0),Eq(x(t) - 2*y(t) + z(t) + Derivative(y(t), t),0),Eq(y(t) - 3*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{t} - C_{2} e^{3 t} + C_{3} e^{4 t}, \ y{\left (t \right )} = 2 C_{1} e^{t} - C_{3} e^{4 t}, \ z{\left (t \right )} = C_{1} e^{t} + C_{2} e^{3 t} + C_{3} e^{4 t}\right ] \]