14.13.10 problem 10

Internal problem ID [3819]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 06:57:44 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.141 (sec). Leaf size: 18

ode:=[diff(x__1(t),t) = 2*x__1(t)+5*x__2(t), diff(x__2(t),t) = -x__1(t)-2*x__2(t)]; 
ic:=[x__1(0) = 0, x__2(0) = 1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= 5 \sin \left (t \right ) \\ x_{2} \left (t \right ) &= \cos \left (t \right )-2 \sin \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 19

ode={D[x1[t],t]==2*x1[t]+5*x2[t],D[x2[t],t]==-x1[t]-2*x2[t]}; 
ic={x1[0]==0,x2[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to 5 \sin (t)\\ \text {x2}(t)&\to \cos (t)-2 \sin (t) \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 31

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

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