76.25.12 problem 12

Internal problem ID [17746]
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 : 12
Date solved : Thursday, March 13, 2025 at 10:48:19 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 36

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

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

✓ Sympy. Time used: 0.134 (sec). Leaf size: 44

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)/2 + x__2(t) + 3*x__3(t)/2 + Derivative(x__1(t), t),0),Eq(-3*x__1(t)/2 + 2*x__2(t) + 3*x__3(t)/2 + Derivative(x__2(t), t),0),Eq(2*x__1(t) - 2*x__2(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)
 

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