20.20.30 problem 34

Internal problem ID [3920]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 34
Date solved : Tuesday, March 04, 2025 at 05:19:28 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )+4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=8 x_{1} \left (t \right )+x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = -3*x__1(t)+4*x__2(t), diff(x__2(t),t) = 8*x__1(t)+x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{-7 t}+c_{2} {\mathrm e}^{5 t} \\ x_{2} \left (t \right ) &= -c_{1} {\mathrm e}^{-7 t}+2 c_{2} {\mathrm e}^{5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 71

ode={D[x1[t],t]==-3*x1[t]+4*x2[t],D[x2[t],t]==8*x1[t]+1*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-7 t} \left (c_1 \left (e^{12 t}+2\right )+c_2 \left (e^{12 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-7 t} \left (2 c_1 \left (e^{12 t}-1\right )+c_2 \left (2 e^{12 t}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.099 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(3*x__1(t) - 4*x__2(t) + Derivative(x__1(t), t),0),Eq(-8*x__1(t) - x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{1} e^{- 7 t} + \frac {C_{2} e^{5 t}}{2}, \ x^{2}{\left (t \right )} = C_{1} e^{- 7 t} + C_{2} e^{5 t}\right ] \]