76.8.12 problem 12

Internal problem ID [17422]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 12
Date solved : Thursday, March 13, 2025 at 10:08:06 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.079 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 56

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

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

✓ Sympy. Time used: 0.119 (sec). Leaf size: 48

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

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