87.19.15 problem 29 and 30

Internal problem ID [23685]
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 : 29 and 30
Date solved : Thursday, October 02, 2025 at 09:44:14 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-10 x_{1} \left (t \right )+x_{2} \left (t \right )+7 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-9 x_{1} \left (t \right )+4 x_{2} \left (t \right )+5 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-17 x_{1} \left (t \right )+x_{2} \left (t \right )+12 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right )&=5 \\ x_{2} \left (0\right )&=2 \\ x_{3} \left (0\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 61

ode:=[diff(x__1(t),t) = -10*x__1(t)+x__2(t)+7*x__3(t), diff(x__2(t),t) = -9*x__1(t)+4*x__2(t)+5*x__3(t), diff(x__3(t),t) = -17*x__1(t)+x__2(t)+12*x__3(t)]; 
ic:=[x__1(0) = 5, x__2(0) = 2, x__3(0) = -2]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= -87 \,{\mathrm e}^{2 t}+10 \,{\mathrm e}^{3 t}+82 \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= -29 \,{\mathrm e}^{2 t}-10 \,{\mathrm e}^{3 t}+41 \,{\mathrm e}^{t} \\ x_{3} \left (t \right ) &= -145 \,{\mathrm e}^{2 t}+20 \,{\mathrm e}^{3 t}+123 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 65

ode={D[x1[t],t]==-10*x1[t]+x2[t]+7*x3[t],D[x2[t],t]==-9*x1[t]+4*x2[t]+5*x3[t],D[x3[t],t]==-17*x1[t]+x2[t]+12*x3[t]}; 
ic={x1[0]==5,x2[0]==2,x3[0]==-2}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to e^t \left (-87 e^t+10 e^{2 t}+82\right )\\ \text {x2}(t)&\to e^t \left (-29 e^t-10 e^{2 t}+41\right )\\ \text {x3}(t)&\to e^t \left (-145 e^t+20 e^{2 t}+123\right ) \end{align*}

✓ Sympy. Time used: 0.159 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
x3 = Function("x3") 
ode=[Eq(10*x1(t) - x2(t) - 7*x3(t) + Derivative(x1(t), t),0),Eq(9*x1(t) - 4*x2(t) - 5*x3(t) + Derivative(x2(t), t),0),Eq(17*x1(t) - x2(t) - 12*x3(t) + Derivative(x3(t), t),0)] 
ics = {x1(0): 5, x2(0): 2, x3(0): -2} 
dsolve(ode,func=[x1(t),x2(t),x3(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = 10 e^{3 t} - 87 e^{2 t} + 82 e^{t}, \ x_{2}{\left (t \right )} = - 10 e^{3 t} - 29 e^{2 t} + 41 e^{t}, \ x_{3}{\left (t \right )} = 20 e^{3 t} - 145 e^{2 t} + 123 e^{t}\right ] \]