87.21.17 problem 17

Internal problem ID [23731]
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 : 17
Date solved : Thursday, October 02, 2025 at 09:44:40 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=3 x+2 y \left (t \right )+2 z \left (t \right )\\ y^{\prime }\left (t \right )&=x+4 y \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=-2 x-4 y \left (t \right )-z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.077 (sec). Leaf size: 48

ode:=[diff(x(t),t) = 3*x(t)+2*y(t)+2*z(t), diff(y(t),t) = x(t)+4*y(t)+z(t), diff(z(t),t) = -2*x(t)-4*y(t)-z(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 108

ode={D[x[t],t]==3*x[t]+2*y[t]+2*z[t],D[y[t],t]==x[t]+4*y[t]+z[t],D[z[t],t]==-2*x[t]-4*y[t]-z[t] }; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.082 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-3*x(t) - 2*y(t) - 2*z(t) + Derivative(x(t), t),0),Eq(-x(t) - 4*y(t) - z(t) + Derivative(y(t), t),0),Eq(2*x(t) + 4*y(t) + z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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