10.16.12 problem 12

Internal problem ID [1412]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 12
Date solved : Tuesday, March 04, 2025 at 12:35:26 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {4 x_{1} \left (t \right )}{5}+2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+\frac {6 x_{2} \left (t \right )}{5} \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 45

ode:=[diff(x__1(t),t) = -4/5*x__1(t)+2*x__2(t), diff(x__2(t),t) = -x__1(t)+6/5*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 56

ode={D[ x1[t],t]==-4/5*x1[t]+2*x2[t],D[ x2[t],t]==-1*x1[t]+6/5*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.116 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(4*x__1(t)/5 - 2*x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) - 6*x__2(t)/5 + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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