28.4.32 problem 7.32

Internal problem ID [4564]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.32
Date solved : Sunday, March 30, 2025 at 03:26:22 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=5 x_{1} \left (t \right )+3 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-3 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.137 (sec). Leaf size: 28

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (-3 t +1\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (-9 t +6\right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 30

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

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

✓ Sympy. Time used: 0.104 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-5*x__1(t) - 3*x__2(t) + Derivative(x__1(t), t),0),Eq(3*x__1(t) + x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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