87.21.29 problem 29

Internal problem ID [23743]
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 : 29
Date solved : Thursday, October 02, 2025 at 09:44:49 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=-10 x+y \left (t \right )+7 z \left (t \right )\\ y^{\prime }\left (t \right )&=-9 x+4 y \left (t \right )+5 z \left (t \right )\\ z^{\prime }\left (t \right )&=-17 x+y \left (t \right )+12 z \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=6 \\ y \left (0\right )&=1 \\ z \left (0\right )&=10 \\ \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 55

ode:=[diff(x(t),t) = -10*x(t)+y(t)+7*z(t), diff(y(t),t) = -9*x(t)+4*y(t)+5*z(t), diff(z(t),t) = -17*x(t)+y(t)+12*z(t)]; 
ic:=[x(0) = 6, y(0) = 1, z(0) = 10]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{2 t}+{\mathrm e}^{3 t}+2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= {\mathrm e}^{2 t}-{\mathrm e}^{3 t}+{\mathrm e}^{t} \\ z \left (t \right ) &= 5 \,{\mathrm e}^{2 t}+2 \,{\mathrm e}^{3 t}+3 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 61

ode={D[x[t],t]==-10*x[t]+y[t]+7*z[t],D[y[t],t]==-9*x[t]+4*y[t]+5*z[t],D[z[t],t]==-17*x[t]+y[t]+12*z[t]}; 
ic={x[0]==6,y[0]==1,z[0]==10}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^t \left (3 e^t+e^{2 t}+2\right )\\ y(t)&\to e^t+e^{2 t}-e^{3 t}\\ z(t)&\to e^t \left (5 e^t+2 e^{2 t}+3\right ) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(10*x(t) - y(t) - 7*z(t) + Derivative(x(t), t),0),Eq(9*x(t) - 4*y(t) - 5*z(t) + Derivative(y(t), t),0),Eq(17*x(t) - y(t) - 12*z(t) + Derivative(z(t), t),0)] 
ics = {x(0): 6, y(0): 1, z(0): 10} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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