70.25.10 problem 10

Internal problem ID [19109]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.3 (Homogeneous Linear Systems with Constant Coefficients). Problems at page 408
Problem number : 10
Date solved : Thursday, October 02, 2025 at 03:38:03 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right )&=7 \\ x_{2} \left (0\right )&=5 \\ x_{3} \left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 61

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

\begin{align*} x_{1} \left (t \right ) &= 6 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{-t}-2 \,{\mathrm e}^{4 t} \\ x_{2} \left (t \right ) &= 12 \,{\mathrm e}^{t}-6 \,{\mathrm e}^{-t}-{\mathrm e}^{4 t} \\ x_{3} \left (t \right ) &= -6 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{-t}+8 \,{\mathrm e}^{4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 71

ode={D[x1[t],t]==0*x1[t]+0*x2[t]-1*x3[t],D[x2[t],t]==2*x1[t]+0*x2[t]+0*x3[t],D[x3[t],t]==-1*x1[t]+2*x2[t]+4*x3[t]}; 
ic={x1[0]==7,x2[0]==5,x3[0]==5}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to 3 e^{-t}+6 e^t-2 e^{4 t}\\ \text {x2}(t)&\to -6 e^{-t}+12 e^t-e^{4 t}\\ \text {x3}(t)&\to 3 e^{-t}-6 e^t+8 e^{4 t} \end{align*}

✓ Sympy. Time used: 0.077 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(x__3(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) + Derivative(x__2(t), t),0),Eq(x__1(t) - 2*x__2(t) - 4*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{- t} - C_{2} e^{t} - \frac {C_{3} e^{4 t}}{4}, \ x^{2}{\left (t \right )} = - 2 C_{1} e^{- t} - 2 C_{2} e^{t} - \frac {C_{3} e^{4 t}}{8}, \ x^{3}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t} + C_{3} e^{4 t}\right ] \]