7.25.12 problem 12

Internal problem ID [632]
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 : 12
Date solved : Saturday, March 29, 2025 at 05:01:09 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.130 (sec). Leaf size: 58

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (2 t \right ) c_1 +\cos \left (2 t \right ) c_2 \right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (\sin \left (2 t \right ) c_1 -2 \sin \left (2 t \right ) c_2 +2 \cos \left (2 t \right ) c_1 +\cos \left (2 t \right ) c_2 \right )}{5} \\ \end{align*}

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

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

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{2 t} (2 c_1 \cos (2 t)-(c_1+5 c_2) \sin (2 t)) \\ \text {x2}(t)\to \frac {1}{2} e^{2 t} (2 c_2 \cos (2 t)+(c_1+c_2) \sin (2 t)) \\ \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 60

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

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