48.5.11 problem Problem 5.12

Internal problem ID [7576]
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 5.12
Date solved : Sunday, March 30, 2025 at 12:15:52 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )-18 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-9 x_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.131 (sec). Leaf size: 28

ode:=[diff(x__1(t),t) = 3*x__1(t)-18*x__2(t), diff(x__2(t),t) = 2*x__1(t)-9*x__2(t)]; 
ic:=x__1(0) = 2x__2(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (-6 t +2\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (-36 t +18\right )}{18} \\ \end{align*}

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

ode={D[ x1[t],t]==3*x1[t]-18*x2[t],D[ x2[t],t]==2*x1[t]-9*x2[t]}; 
ic={x1[0]==2,x2[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-3 t} (2-6 t) \\ \text {x2}(t)\to e^{-3 t} (1-2 t) \\ \end{align*}

✓ Sympy. Time used: 0.135 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) + 18*x__2(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) + 9*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = 6 C_{1} t e^{- 3 t} + \left (C_{1} + 6 C_{2}\right ) e^{- 3 t}, \ x^{2}{\left (t \right )} = 2 C_{1} t e^{- 3 t} + 2 C_{2} e^{- 3 t}\right ] \]