87.19.14 problem 25 and 27

Internal problem ID [23684]
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 149
Problem number : 25 and 27
Date solved : Thursday, October 02, 2025 at 09:44:14 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=5 x_{1} \left (t \right )+2 x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+2 x_{2} \left (t \right )-4 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-4 x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right )&=2 \\ x_{2} \left (0\right )&=1 \\ x_{3} \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.070 (sec). Leaf size: 21

ode:=[diff(x__1(t),t) = 5*x__1(t)+2*x__2(t)+2*x__3(t), diff(x__2(t),t) = 2*x__1(t)+2*x__2(t)-4*x__3(t), diff(x__3(t),t) = 2*x__1(t)-4*x__2(t)+2*x__3(t)]; 
ic:=[x__1(0) = 2, x__2(0) = 1, x__3(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= 2 \,{\mathrm e}^{6 t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{6 t} \\ x_{3} \left (t \right ) &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 24

ode={D[x1[t],t]==5*x1[t]+2*x2[t]+2*x3[t],D[x2[t],t]==2*x1[t]+2*x2[t]-4*x3[t],D[x3[t],t]==2*x1[t]-4*x2[t]+2*x3[t]}; 
ic={x1[0]==2,x2[0]==1,x3[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to 2 e^{6 t}\\ \text {x2}(t)&\to e^{6 t}\\ \text {x3}(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.146 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
x3 = Function("x3") 
ode=[Eq(-5*x1(t) - 2*x2(t) - 2*x3(t) + Derivative(x1(t), t),0),Eq(-2*x1(t) - 2*x2(t) + 4*x3(t) + Derivative(x2(t), t),0),Eq(-2*x1(t) + 4*x2(t) - 2*x3(t) + Derivative(x3(t), t),0)] 
ics = {x1(0): 2, x2(0): 1, x3(0): 0} 
dsolve(ode,func=[x1(t),x2(t),x3(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = 2 e^{6 t}, \ x_{2}{\left (t \right )} = e^{6 t}, \ x_{3}{\left (t \right )} = 0\right ] \]