76.27.10 problem 10

Internal problem ID [17785]
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 6.5 (Fundamental Matrices and the Exponential of a Matrix). Problems at page 430
Problem number : 10
Date solved : Monday, March 31, 2025 at 04:27:39 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.114 (sec). Leaf size: 30

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

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-\frac {3 t}{2}} \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{t}-4 c_2 \,{\mathrm e}^{-\frac {3 t}{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 83

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

\begin{align*} \text {x1}(t)\to \frac {1}{5} e^{-3 t/2} \left (c_1 \left (4 e^{5 t/2}+1\right )+c_2 \left (e^{5 t/2}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{5} e^{-3 t/2} \left (4 c_1 \left (e^{5 t/2}-1\right )+c_2 \left (e^{5 t/2}+4\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.087 (sec). Leaf size: 32

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

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