14.20.10 problem 10

Internal problem ID [3900]
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 : 10
Date solved : Tuesday, September 30, 2025 at 06:58:48 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )-10 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=5 x_{1} \left (t \right )+11 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.149 (sec). Leaf size: 47

ode:=[diff(x__1(t),t) = -3*x__1(t)-10*x__2(t), diff(x__2(t),t) = 5*x__1(t)+11*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{4 t} \left (\sin \left (t \right ) c_1 +\cos \left (t \right ) c_2 \right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{4 t} \left (7 \sin \left (t \right ) c_1 -\sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 +7 \cos \left (t \right ) c_2 \right )}{10} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 57

ode={D[x1[t],t]==-3*x1[t]-10*x2[t],D[x2[t],t]==5*x1[t]+11*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to e^{4 t} (c_1 \cos (t)-(7 c_1+10 c_2) \sin (t))\\ \text {x2}(t)&\to e^{4 t} (c_2 \cos (t)+(5 c_1+7 c_2) \sin (t)) \end{align*}

✓ Sympy. Time used: 0.074 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(3*x__1(t) + 10*x__2(t) + Derivative(x__1(t), t),0),Eq(-5*x__1(t) - 11*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \left (\frac {C_{1}}{5} - \frac {7 C_{2}}{5}\right ) e^{4 t} \sin {\left (t \right )} - \left (\frac {7 C_{1}}{5} + \frac {C_{2}}{5}\right ) e^{4 t} \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = C_{1} e^{4 t} \cos {\left (t \right )} - C_{2} e^{4 t} \sin {\left (t \right )}\right ] \]