87.21.22 problem 22

Internal problem ID [23736]
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 : 22
Date solved : Thursday, October 02, 2025 at 09:44:44 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.075 (sec). Leaf size: 86

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

\begin{align*} h \left (t \right ) &= c_2 \,{\mathrm e}^{4 t}+c_3 \,{\mathrm e}^{t} \\ w \left (t \right ) &= c_2 \,{\mathrm e}^{4 t}+c_3 \,{\mathrm e}^{t}+{\mathrm e}^{t} c_1 \\ x \left (t \right ) &= c_4 \,{\mathrm e}^{4 t}+c_5 \,{\mathrm e}^{2 t} \\ y \left (t \right ) &= c_4 \,{\mathrm e}^{4 t}-c_5 \,{\mathrm e}^{2 t} \\ z \left (t \right ) &= c_2 \,{\mathrm e}^{4 t}-2 c_3 \,{\mathrm e}^{t}-{\mathrm e}^{t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 184

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

\begin{align*} x(t)&\to \frac {1}{2} e^{2 t} \left (c_1 \left (e^{2 t}+1\right )+c_2 \left (e^{2 t}-1\right )\right )\\ y(t)&\to \frac {1}{2} e^{2 t} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )\right )\\ h(t)&\to \frac {1}{3} e^t \left (c_3 \left (e^{3 t}+2\right )+(c_4+c_5) \left (e^{3 t}-1\right )\right )\\ w(t)&\to \frac {1}{3} e^t \left (c_3 \left (e^{3 t}-1\right )+c_4 \left (e^{3 t}+2\right )+c_5 \left (e^{3 t}-1\right )\right )\\ z(t)&\to \frac {1}{3} e^t \left (c_3 \left (e^{3 t}-1\right )+c_4 \left (e^{3 t}-1\right )+c_5 \left (e^{3 t}+2\right )\right ) \end{align*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 73

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

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