87.21.21 problem 21

Internal problem ID [23735]
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 : 21
Date solved : Thursday, October 02, 2025 at 09:44:43 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=x+2 y \left (t \right )+z \left (t \right )-w \left (t \right )\\ y^{\prime }\left (t \right )&=-y \left (t \right )+2 z \left (t \right )+2 w \left (t \right )\\ z^{\prime }\left (t \right )&=2 y \left (t \right )+2 z \left (t \right )+2 w \left (t \right )\\ w^{\prime }\left (t \right )&=-3 y \left (t \right )-6 z \left (t \right )-6 w \left (t \right ) \end{align*}

✓ Maple. Time used: 0.087 (sec). Leaf size: 68

ode:=[diff(x(t),t) = x(t)+2*y(t)+z(t)-w(t), diff(y(t),t) = -y(t)+2*z(t)+2*w(t), diff(z(t),t) = 2*y(t)+2*z(t)+2*w(t), diff(w(t),t) = -3*y(t)-6*z(t)-6*w(t)]; 
dsolve(ode);
 

\begin{align*} w \left (t \right ) &= c_3 +c_4 \,{\mathrm e}^{-3 t} \\ x \left (t \right ) &= c_1 \,{\mathrm e}^{t}+2 c_3 -\frac {{\mathrm e}^{-2 t} c_2}{2}+\frac {3 c_4 \,{\mathrm e}^{-3 t}}{4} \\ y \left (t \right ) &= -c_4 \,{\mathrm e}^{-3 t}+{\mathrm e}^{-2 t} c_2 \\ z \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} c_2}{2}-c_3 \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 205

ode={D[x[t],t]==x[t]+2*y[t]+z[t]-w[t],D[y[t],t]==-y[t]+2*z[t]+2*w[t],D[z[t],t]==2*y[t]+2*z[t]+2*w[t],D[w[t],t]==-3*y[t]-6*z[t]-6*w[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t],w[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} w(t)&\to e^{-3 t} \left (-\left (c_1 \left (e^{3 t}-2\right )\right )-(c_3+2 c_4) \left (e^{3 t}-1\right )\right )\\ x(t)&\to \frac {1}{4} e^{-3 t} \left (c_1 \left (-4 e^t-8 e^{3 t}+6 e^{4 t}+6\right )-4 (c_3+c_4) e^t-8 (c_3+2 c_4) e^{3 t}+(4 c_2+9 c_3+14 c_4) e^{4 t}+3 (c_3+2 c_4)\right )\\ y(t)&\to e^{-3 t} \left (2 c_1 \left (e^t-1\right )+c_3 \left (2 e^t-1\right )+2 c_4 \left (e^t-1\right )\right )\\ z(t)&\to c_1 \left (-e^{-2 t}\right )-c_3 e^{-2 t}-c_4 e^{-2 t}+c_1+c_3+2 c_4 \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
w = Function("w") 
ode=[Eq(w(t) - x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(-2*w(t) + y(t) - 2*z(t) + Derivative(y(t), t),0),Eq(-2*w(t) - 2*y(t) - 2*z(t) + Derivative(z(t), t),0),Eq(6*w(t) - 3*y(t) + 6*z(t) + Derivative(w(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t),w(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} - \frac {5 C_{2} e^{t}}{3} - \frac {5 C_{3} t e^{t}}{3} + \frac {5 C_{4} e^{- 6 t}}{21}, \ y{\left (t \right )} = - \frac {C_{3} e^{t}}{3} - \frac {C_{4} e^{- 6 t}}{3}, \ z{\left (t \right )} = - C_{1} - \frac {4 C_{3} e^{t}}{3} - \frac {C_{4} e^{- 6 t}}{6}, \ w{\left (t \right )} = C_{1} + C_{3} e^{t} + C_{4} e^{- 6 t}\right ] \]