48.5.8 problem Problem 5.9

Internal problem ID [7573]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 5. Systems of First Order Differential Equations. Section 5.11 Problems. Page 360
Problem number : Problem 5.9
Date solved : Sunday, March 30, 2025 at 12:15:48 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.124 (sec). Leaf size: 33

ode:=[diff(x__1(t),t) = x__1(t)-2*x__2(t), diff(x__2(t),t) = 3*x__1(t)-4*x__2(t)]; 
ic:=x__1(0) = 1x__2(0) = 0; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 33

ode={D[ x1[t],t]==x1[t]-2*x2[t],D[ x2[t],t]==3*x1[t]-4*x2[t]}; 
ic={x1[0]==1,x2[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-2 t} \left (3 e^t-2\right ) \\ \text {x2}(t)\to 3 e^{-2 t} \left (e^t-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.131 (sec). Leaf size: 31

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) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) + 4*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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