76.25.9 problem 9

Internal problem ID [17743]
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 : 9
Date solved : Thursday, March 13, 2025 at 10:48:16 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 11\\ x_{2} \left (0\right ) = 1\\ x_{3} \left (0\right ) = 5 \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 31

ode:=[diff(x__1(t),t) = x__1(t)+x__2(t)+2*x__3(t), diff(x__2(t),t) = 2*x__2(t)+2*x__3(t), diff(x__3(t),t) = -x__1(t)+x__2(t)+3*x__3(t)]; 
ic:=x__1(0) = 11x__2(0) = 1x__3(0) = 5; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= 11 \,{\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= 11 \,{\mathrm e}^{2 t}-10 \,{\mathrm e}^{t} \\ x_{3} \left (t \right ) &= 5 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 34

ode={D[x1[t],t]==1*x1[t]+1*x2[t]+2*x3[t],D[x2[t],t]==0*x1[t]+2*x2[t]+2*x3[t],D[x3[t],t]==-1*x1[t]+1*x2[t]+3*x3[t]}; 
ic={x1[0]==11,x2[0]==1,x3[0]==5}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to 11 e^{2 t} \\ \text {x2}(t)\to e^t \left (11 e^t-10\right ) \\ \text {x3}(t)\to 5 e^t \\ \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 54

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

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