85.87.2 problem 2

Internal problem ID [23026]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. B Exercises at page 491
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:17:35 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+5 y \left (t \right )+3 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=3 x \left (t \right )+9 y \left (t \right )+5 z \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=-2 \\ y \left (0\right )&=-1 \\ z \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 13

ode:=[diff(x(t),t) = x(t)+y(t)+z(t), diff(y(t),t) = 2*x(t)+5*y(t)+3*z(t), diff(z(t),t) = 3*x(t)+9*y(t)+5*z(t)]; 
ic:=[x(0) = -2, y(0) = -1, z(0) = 3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= -2 \\ y \left (t \right ) &= -1 \\ z \left (t \right ) &= 3 \\ \end{align*}

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 14

ode={D[x[t],{t,1}]==x[t]+y[t]+z[t],D[y[t],{t,1}]==2*x[t]+5*y[t]+3*z[t],D[z[t],t]==3*x[t]+9*y[t]+5*z[t]}; 
ic={x[0]==-2,y[0]==-1,z[0]==3}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -2\\ y(t)&\to -1\\ z(t)&\to 3 \end{align*}

✓ Sympy. Time used: 0.226 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x(t) - y(t) - z(t) + Derivative(x(t), t),0),Eq(-2*x(t) - 5*y(t) - 3*z(t) + Derivative(y(t), t),0),Eq(-3*x(t) - 9*y(t) - 5*z(t) + Derivative(z(t), t),0)] 
ics = {x(0): -2, y(0): -1, z(0): 3} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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