20.20.32 problem 36

Internal problem ID [3922]
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 : 36
Date solved : Tuesday, March 04, 2025 at 05:19:30 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=5 x_{1} \left (t \right )+9 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 59

ode:=[diff(x__1(t),t) = 5*x__1(t)+9*x__2(t), diff(x__2(t),t) = -2*x__1(t)-x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 67

ode={D[x1[t],t]==5*x1[t]+9*x2[t],D[x2[t],t]==-2*x1[t]-1*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.123 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-5*x__1(t) - 9*x__2(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) + x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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