20.16.18 problem 18

Internal problem ID [3851]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 18
Date solved : Tuesday, March 04, 2025 at 05:18:03 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )-6 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{1} \left (t \right )+5 x_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 44

ode:=[diff(x__1(t),t) = -x__1(t)-6*x__2(t), diff(x__2(t),t) = 3*x__1(t)+5*x__2(t)]; 
ic:=x__1(0) = 2x__2(0) = 2; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (-6 \sin \left (3 t \right )+2 \cos \left (3 t \right )\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (-4 \cos \left (3 t \right )-8 \sin \left (3 t \right )\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 44

ode={D[x1[t],t]==-x1[t]-6*x2[t],D[x2[t],t]==3*x1[t]+5*x2[t]}; 
ic={x1[0]==2,x2[0]==2}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.112 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(x__1(t) + 6*x__2(t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) - 5*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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