76.25.11 problem 11

Internal problem ID [17745]
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 : 11
Date solved : Thursday, March 13, 2025 at 10:48:18 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 115

ode:=[diff(x__1(t),t) = x__1(t)+3*x__3(t), diff(x__2(t),t) = -2*x__2(t), diff(x__3(t),t) = 3*x__1(t)-x__3(t)]; 
ic:=x__1(0) = 2x__2(0) = -1x__3(0) = -2; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= \left (1-\frac {\sqrt {10}}{5}\right ) {\mathrm e}^{\sqrt {10}\, t}+\left (1+\frac {\sqrt {10}}{5}\right ) {\mathrm e}^{-\sqrt {10}\, t} \\ x_{2} \left (t \right ) &= -{\mathrm e}^{-2 t} \\ x_{3} \left (t \right ) &= \frac {\left (1-\frac {\sqrt {10}}{5}\right ) \sqrt {10}\, {\mathrm e}^{\sqrt {10}\, t}}{3}-\frac {\left (1+\frac {\sqrt {10}}{5}\right ) \sqrt {10}\, {\mathrm e}^{-\sqrt {10}\, t}}{3}-\frac {\left (1-\frac {\sqrt {10}}{5}\right ) {\mathrm e}^{\sqrt {10}\, t}}{3}-\frac {\left (1+\frac {\sqrt {10}}{5}\right ) {\mathrm e}^{-\sqrt {10}\, t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 32

ode={D[x1[t],t]==-1*x1[t]+0*x2[t]+3*x3[t],D[x2[t],t]==0*x1[t]-2*x2[t]+0*x3[t],D[x3[t],t]==3*x1[t]+0*x2[t]-1*x3[t]}; 
ic={x1[0]==2,x2[0]==-1,x3[0]==-2}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to 2 e^{-4 t} \\ \text {x3}(t)\to -2 e^{-4 t} \\ \text {x2}(t)\to -e^{-2 t} \\ \end{align*}

✓ Sympy. Time used: 0.193 (sec). Leaf size: 70

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-x__1(t) - 3*x__3(t) + Derivative(x__1(t), t),0),Eq(2*x__2(t) + Derivative(x__2(t), t),0),Eq(-3*x__1(t) + x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} \left (1 - \sqrt {10}\right ) e^{- \sqrt {10} t}}{3} + \frac {C_{2} \left (1 + \sqrt {10}\right ) e^{\sqrt {10} t}}{3}, \ x^{2}{\left (t \right )} = C_{3} e^{- 2 t}, \ x^{3}{\left (t \right )} = C_{1} e^{- \sqrt {10} t} + C_{2} e^{\sqrt {10} t}\right ] \]