7.25.11 problem 11

Internal problem ID [631]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 11
Date solved : Saturday, March 29, 2025 at 05:01:07 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 4 \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 23

ode:=[diff(x__1(t),t) = x__1(t)-2*x__2(t), diff(x__2(t),t) = 2*x__1(t)+x__2(t)]; 
ic:=x__1(0) = 0x__2(0) = 4; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 26

ode={D[x1[t],t]==x1[t]-2*x2[t],D[x2[t],t]==2*x1[t]+x2[t]}; 
ic={x1[0]==0,x2[0]==4}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to -4 e^t \sin (2 t) \\ \text {x2}(t)\to 4 e^t \cos (2 t) \\ \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) + 2*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)
 

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